journal
J. Am. Ceram. Soc., 83 [9] 2117– 46 (2000)
Origins and Applications of London Dispersion
Forces and Hamaker Constants in Ceramics
Roger H. French*,**
Central Research, DuPont Company, Wilmington, Delaware 19880; and Materials Science Department,
University of Pennsylvania, Philadelphia, Pennsylvania 19104
The London dispersion forces, along with the Debye and
Keesom forces, constitute the long-range van der Waals forces.
London’s and Hamaker’s work on the point-to-point dispersion interaction and Lifshitz’s development of the continuum
theory of dispersion are the foundations of our understanding
of dispersion forces. Dispersion forces are present for all
materials and are intrinsically related to the optical properties
and the underlying interband electronic structures of materials. The force law scaling constant of the dispersion force,
known as the Hamaker constant, can be determined from
spectral or parametric optical properties of materials, combined with knowledge of the configuration of the materials.
With recent access to new experimental and ab initio tools for
determination of optical properties of materials, dispersion
force research has new opportunities for detailed studies.
Opportunities include development of improved index approximations and parametric representations of the optical properties for estimation of Hamaker constants. Expanded databases of London dispersion spectra of materials will permit
accurate estimation of both nonretarded and retarded dispersion forces in complex configurations. Development of solutions for generalized multilayer configurations of materials are
needed for the treatment of more-complex problems, such as
graded interfaces. Dispersion forces can play a critical role in
materials applications. Typically, they are a component with
other forces in a force balance, and it is this balance that
dictates the resulting behavior. The ubiquitous nature of the
London dispersion forces makes them a factor in a wide
spectrum of problems; they have been in evidence since the
pioneering work of Young and Laplace on wetting, contact
angles, and surface energies. Additional applications include
the interparticle forces that can be measured by direct techniques, such as atomic force microscopy. London dispersion
forces are important in both adhesion and in sintering, where
the detailed shape at the crack tip and at the sintering neck can
be controlled by the dispersion forces. Dispersion forces have
—D. R. Clarke
Manuscript No. 188760. Received February 1, 2000; approved June 5, 2000.
*Member, American Ceramic Society.
**Fellow, American Ceramic Society.
Article Table of Contents
Introduction 2118
Overview 2118
History 2118
Structure of This Article 2119
Origins of Dispersion Forces 2119
van der Waals Forces 2119
Lifshitz Theory and Electronic Structure and
Bonding 2121
Full Spectral Nonretarded Hamaker Constant ANR
from Spectral Optical Properties 2125
Methods for Estimating Hamaker
Constant ANR 2128
Retardation of Dispersion Forces and the Retarded
Hamaker Constant AR 2129
Applications 2129
Classes of Materials 2130
Surface Energies, Wetting, and Contact
Angles 2130
Balance of Forces 2132
Colloidal Systems 2132
Interparticle Forces 2132
Adhesion and Sintering 2133
Equilibrium Intergranular and Surficial
Films 2135
Retardation of Dispersion Forces 2139
Conclusions 2141
Acknowledgements 2142
References 2142
an important role in the properties of numerous ceramics that
contain intergranular films, and here the opportunity exists for
the development of an integrated understanding of intergranular films that encompasses dispersion forces, segregation,
multilayer adsorption, and structure. The intrinsic length scale
at which there is a transition from the continuum perspective
(dispersion forces) to the atomistic perspective (encompassing
interatomic bonds) is critical in many materials problems, and
the relationship of dispersion forces and intergranular films
may represent an important opportunity to probe this topic.
Centennial Feature
2118
Journal of the American Ceramic Society—French
The London dispersion force is retarded at large separations,
where the transit time of the electromagnetic interaction must
be considered explicitly. Novel phenomena, such as equilibrium surficial films and bimodal wetting/dewetting, can result
in materials systems when the characteristic wavelengths of
the interatomic bonds and the physical interlayer thicknesses
lead to a change in the sign of the dispersion force. Use of these
novel phenomena in future materials applications provides
interesting opportunities in materials design.
I.
Introduction
T
origins and applications of the van der Waals (vdW) force
and its major component, the London dispersion (LD) force,
have been of continuous interest for more than 200 years, spanning
virtually all areas of science from mathematics to physics, chemistry, and biology. Dispersion forces are manifest throughout the
world. Examples are the wetting of solids by liquids, capillarity of
liquids, structure of biological cell walls, flocculation of colloidal
suspensions, adhesion and fracture of materials, and numerous
other microscopic and macroscopic phenomena.
HE
(1) Overview
Research in dispersion forces has elucidated their role in many
practical application areas and phenomena. Our fundamental
understanding of dispersion forces mostly derives from the work
of Lifshitz, and, in the 50 years since his work, we have expanded
the foundation of dispersion force science and advanced its
application to real world problems, such as wetting and colloidal
processing.
Improved spectral data, numerical methods, and advances in
direct force measurements have occurred during the past 15 years.
The spectral data allow the direct determination of the Hamaker
constant (A) for the dispersion force from first-principles Lifshitz
theory. This, combined with the ability to directly measure these
forces using atomic force microscopy, has produced a renaissance
of research activity. Dispersion force theory is now applied to such
areas as sintering and crack propagation and to the role of
intergranular glassy films in ceramics. Our improved understanding of dispersion forces has advanced our understanding of
room-temperature colloidal processing of ceramics and hightemperature processing of ceramics exhibiting intergranular films.
We also understand better the design and performance of structural
and electronic ceramics from viscous-sintered systems exhibiting
intergranular films.
Dispersion forces have such wide application that they are
intrinsically interdisciplinary in nature, spanning scientific and
industrial applications and all fields of science. As advances have
occurred in one area of science, they have enabled advances in
other apparently unrelated fields that are united only by the
underlying role played by dispersion forces. It is for this reason
that we undertake a detailed view of the history, current state, and
future directions of our understanding of the LD forces.
(2) History
One of the earliest references to the wetting of solids by liquids
can be found in the observations of Taylor and of Hauksbee, who,
in the early 1700s, qualitatively described the shape of a fluid
interface that wets a vertical plane.1 In the early 1800s, many
scientists were interested in the behavior of liquids and, in
particular, what aspects of liquids are responsible for their wetting,
capillarity, and the various contact angles that are apparent when
alternate combinations of substrates and liquids are examined.
Among the first works that considered phenomena in which vdW
forces can play a critical role was by Young,2 in 1805, in his “An
Essay on the Cohesion of Fluids,” in which he postulated what has
become known as the Young–Laplace equation for wetting.3 The
discussion of capillary action went on actively, including efforts by
LaPlace, Gauss, Maxwell, and Thomson.4 It was van der Waals5
who, in 1893, developed a thermodynamic theory of capillarity for
application to the behavior of liquids and who established the
Vol. 83, No. 9
minimization of free energy as the criterion for equilibrium in a
liquid– gas system and then applied this to surface tensions. van
der Waals introduced the long-range vdW forces as resulting from
dipole and quadrapolar interactions between molecules that make
up gases, liquids, or solids. van der Waals’ work followed Gibbs’
thermodynamic work on capillarity entitled “On the Equilibrium
of Heterogeneous Substances.”6,7
Surface tension represents the effects of vdW forces of a
material on itself, effectively a self energy, producing a surface
tension at the interface between the material and the gas phase.
London, in 1930,8 –11 using the advances of the then new quantum
mechanics, was able to demonstrate that the vdW forces resulting
from permanent dipoles were not the most universal, because
many materials do not have permanent dipoles, and because
permanent dipole interactions decrease dramatically with increasing temperature. London showed that it is transient-induced
dipoles that result in the dispersion forces; induced dipoles result
from the intrinsic polarizability of the interatomic bonds and the
presence of a propagating electromagnetic field and are long range
and do not disappear at high temperatures. It was from this work
that Hamaker12 extended the understanding of LD forces by
summing the point-by-point interactions among molecules and
producing a measure of the net attraction of two separate bodies.
Hamaker was very active in the 1930s and introduced the dispersion interaction between bodies, leading to the development of the
Hamaker constant A to establish the magnitude of the dispersion
forces.
Even with much progress through the 1930s, the fundamental
relationship between a material’s properties and its LD forces was
not clear. The concept existed of transient dipoles that were
induced in one molecule’s interatomic bonds by the propagating
electromagnetic field produced by the interatomic bonds in adjacent molecules. Because this transient-induced dipole interaction
could be associated with the zero point motion of a quantum
mechanical oscillator whose energy could not be dissipated, it was
realized that the propagating electromagnetic interaction from one
molecule to another was also nondissipative, even as it induced a
polarization interaction. (As shown later, instead of the energy
release associated with a dissipative interaction, this LD interaction results instead from the exchange of virtual photons.)
It was from this perspective—that the interaction propagated as
an induced electromagnetic wave—that eventually brought the
fundamental properties of materials into consideration with the
dispersion forces. For example, Sellmeier,13 in 1872, published a
series of papers entitled “Regarding the Sympathetic Oscillations
Excited in Particles by Oscillations of the Ether and Their
Feedback to the Latter, Particularly as a Means of Explaining
Dispersion and Its Anomalies.” The dispersion referred to here is
the wavelength or frequency dependence of a material’s response
to electromagnetic radiation. Sellmeir’s research was closely
coupled to the development of Maxwell’s wave equations for
electromagnetic radiation. This work by Maxwell and the associated group of scientists, sometimes referred to as the Maxwellians,14 culminated in the 1880s with Heaviside’s15 compact
statement of Maxwell’s equations. It is the close relationship of the
optical properties of materials, the propagation of electromagnetic
waves in materials (as governed by Maxwell’s equations), and the
dispersion of these waves as they travel through a material that ties
the dispersion forces to the optical properties and electronic
structure of materials. However, even by the time of Hamaker’s
1930s work, this connection was not yet made.
In 1948, Casimir16 published a seminal paper on quantum
electrodynamics (QED)17 entitled, “On the Attraction of Two
Perfectly Conducting Plates,” in which he used QED to calculate
the LD force between two metallic plates. This particular dispersion force is now called the Casimir force and has been experimentally confirmed only recently by Lamoreaux.18 Lifshitz then
followed, in 1956,19 in 1960,20 and in 196121 with the general
QED solution (including the effects of retardation) for the LD
forces of two bodies separated by an interlayer material, in which
the optical properties of the bulk materials, as represented by the
dielectric function at optical frequencies, were used to directly
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
calculate the dispersion force. The Lifshitz theory represented a
major advance in the field of dispersion forces because it allowed
the calculation of dispersion forces given known optical properties.
Dispersion force research from the 1960s to the 1980s focused
on determining parametric representations of the optical properties
of materials for use in dispersion force calculations based on
Lifshitz theory. Moreover, much work was done using surface
force apparatus22 to measure intersurface forces between solids.
Knowledge of dispersion forces had a major impact on the theory
of interparticle interactions in colloidal systems and in wetting.
More recently, vacuum ultraviolet (VUV) spectroscopy23 and
valence-electron energy-loss spectroscopy24 have provided the full
spectral optical properties over the wide energy range needed for
Lifshitz calculations, thereby enabling the direct determination of
dispersion forces from optical properties for solid systems. Also,
the use of atomic force microscopy to measure force– distance
curves in numerous systems has led to studies of the dispersion
forces for diverse materials systems.
There have been many excellent articles25,26 and books27–29
that review the origins and determination of the dispersion forces,
calculation of Hamaker constants,30 and the applications of dispersion forces in various fields.31–34
(3) Structure of This Article
vdW forces are long-range forces that can play a critical role in
the interaction of materials. At the same time, they are relatively
weak forces and can be overwhelmed by other shorter-range but
larger forces. It is, therefore, essential to be able to evaluate the
relative role and importance of dispersion forces in problems
involving various classes of materials. We provide an overview of
dispersion forces with an emphasis on applications in ceramics
involving inorganic materials. We also examine some polymeric
and metallic materials that might be used in ceramics processing or
ceramic applications.
We begin with a discussion of the origin of the dispersion forces
in the electronic structure and optical properties of materials. We
next assess the methods for direct determination of Hamaker
constants of the dispersion forces using Lifshitz theory and full
spectral data and then methods of estimating Hamaker constants
from parametric optical data and simple index of refraction
relations. We then look at the effect of retardation of the dispersion
forces at large separations, which result because of the finite speed
of light, and how these retarded forces also can be calculated using
Lifshitz theory.
We review the application of dispersion forces in colloidal
systems, interface energies, and wetting. We next study the
intersurface forces resulting from the dispersion interactions and
the role of dispersion forces in adhesion and sintering. We present
the mechanical and electrical properties of a broad range of
ceramic systems that exhibit intergranular glassy films, mindful of
the importance these films have acquired in the past 15 years for
our understanding of ceramics processing and properties. We then
finish with a discussion of the effects of retardation on wetting,
which can lead to unique wetting conditions. During our presentation, we highlight various topics that represent opportunities for
further research.
II.
Origins of Dispersion Forces
The long-range vdW forces between molecules in materials
result from the interactions of dipoles. We first consider, in Section
II(1)(A), the interactions that produce the vdW forces as simple
point-to-point interactions between dipoles so as to highlight the
nature of the individual interaction. These dipoles can be considered as the individual dipoles present in a molecule or as the
electronic orbitals of an atom or a molecule. To produce a
macroscopic attraction from these dipole interactions, the summation of the dipolar attractions over the atoms or molecules in a bulk
material is required, and this is discussed in Section II(1)(B).
2119
(1) van der Waals Forces
(A) Point-to-Point Dipole Interactions for Molecules: To
determine the vdW force between two molecules, let us first
consider each uncharged molecule as consisting of a permanent or
induced dipole. The vdW forces can be considered as resulting
from three additive terms, the Keesom force, the Debye force, and
the LD force, as shown in Eq. (1).
F vdW ⬇ F Keesom ⫹ F Debye ⫹ F LD
25
(1)
Parsegian has written an intuitive discussion on the fundamental nature of induced dipole interactions and the Debye
induction and LD interactions of the long-range vdW forces.
Beyond the two permanent dipoles that result in the Keesom force,
induced dipoles play a much larger role in the vdW forces. Dipole
moments can be induced in all atoms and molecules and, therefore,
are not restricted to atoms or molecules that have permanent dipole
moments.
(a) Keesom Force—Orientation Effect: In a system of two
permanent dipoles, such as H2O molecules, the interaction of the
dipole’s electric fields results in either an attractive force, when the
dipoles are antiparallel, or a repulsive force, when they are parallel.
This permanent dipole orientation effect results in a force that
Keesom35–38 first discussed in 1912. The Keesom force can be
attractive or repulsive, but vanishes as temperature increases,
because thermally induced motions of the permanent dipoles
disorder their mutual alignment.
(b) Debye Force—Induction Effect: Permanent dipole moments interacting with any atom or molecule can exhibit induced
dipole moments. These result from the intrinsic polarizability of
the multiple electron clouds that constitute the interatomic bonds
of the atom or molecule. This polarizability causes the atom to
respond to an applied electromagnetic field. This can be a static
electric field or, more typically, an electric field that is varying at
the infrared (IR) frequency of a molecular vibration of the
permanent dipole. It also can be the electromagnetic field of a
photon of light that fluctuates at optical frequencies. For time- or
frequency-dependent electromagnetic fields, the polarization response of the electron cloud (i.e., the interatomic bond) also is
frequency dependent. Because the electron cloud has a characteristic frequency that spans the ultraviolet (UV) and visible frequencies, the induced polarization of the electron cloud is responsive to
the imposed electromagnetic field, and it has a much faster
response time to changes in the inducing field. In other words,
electron clouds with induced dipoles have much higher characteristic response frequencies than the very-low-frequency oscillations
or rotations of permanent molecular dipoles.
In 1920, Debye39,40 proposed that a temperature-independent dipolar interaction was possible for a single permanent dipole and its induced
dipoles: The electromagnetic field of a single permanent dipole
induces a dipole moment in the electron cloud of another atom or
molecule. These induced high-frequency electronic dipole moments can couple to the lower-frequency oscillations of the
permanent dipolar molecule. The interaction energy for this
permanent-dipole/induced-dipole interaction results in the Debye
force. This Debye force does not disappear at high temperature and
is always attractive. The Debye force requires the presence of at
least one permanent dipole and, therefore, is not universally
present for all atoms or molecules.
(c) London Force—Dispersion Effect: London8 pointed out
that the Keesom and Debye forces cannot be solely or universally
responsible for the vdW forces, because these forces require the
presence of permanent dipole moments; however, many molecules
that have no permanent dipole moments exhibit long-range vdW
forces. An example London cites are the rare gases that have
negligible dipole and quadropole moments; however, they have
vdW forces that are 100 times larger than can be accounted for by
the Keesom and Debye forces. It was at this point that London
introduced the idea that interatomic bonds in atoms and molecules
can themselves induce dipole moments in nearby interatomic
bonds. The characteristic frequency of interatomic bonds is in the
UV and optical frequency ranges, and the bonds can produce
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Journal of the American Ceramic Society—French
propagating electromagnetic waves or photons. The typical way
that photons are created is when an electron in an excited state
reduces its energy and moves to a lower-energy state, thereby
releasing a propagating electromagnetic wave, or photon, whose
energy is equal to the energy lost by the electronic transition. An
example of this is the production of light that occurs when a
material is heated to a high temperature and then produces radiant
light.
London did not focus on the active production of propagating
electromagnetic waves that result when an electron drops to its
ground state. This active production of light is always associated
with a loss of energy and, therefore, is not the basis of a universal
vdW interaction. Instead, London utilized an idea from the new
quantum mechanics, which stated that an electron, even in its
ground state and at absolute zero temperature, universally exhibits
a zero point motion. This zero point motion of the electron, which
corresponds to the fluctuations of the electron cloud, occurs at the
characteristic frequency of the electron that is in the optical range,
and this ground state energy cannot be dissipated. This zero point
motion of the electron results in a propagating electromagnetic
wave that also cannot be dissipated, and we refer to such waves as
virtual photons so as to distinguish them from energy-dissipating
photons produced from the transition from an excited state to the
ground state.
Similar to the zero point motion of any electron on any atom
producing electromagnetic waves in the form of virtual photons,
these electromagnetic waves and their associated fields can induce
dipole moments in the electron clouds of nearby atoms or
molecules, just as do the electromagnetic fields from permanent
dipoles. It is these induced-dipole/induced-dipole interactions that
result in an attractive force that is the basis of the LD force. The
dispersion force results between individual interatomic bonds that
interact by the exchange of virtual photons, and the dispersion
effect is additive for all the interatomic bonds in atoms or
molecules. The dispersion force is present for all atoms and
molecules and contributes to the vdW force for all materials: It is
a universal contributor to the vdW force.
The LD interaction of two induced dipoles obeys a power-law
relationship with a L⫺6 power for the LD interaction energy (as
shown in Eq. (2)) and a L⫺7 power for the LD force (as shown in
Eq. (3)), where L is the separation distance between the dipoles.
The power-law relationship also exists for the LD interaction of
two materials, but the exponent, as given in Table I, is different
and is discussed in more detail in Section II(1)(B).
E vdW共d兲 ⬀
A Hamaker
Lm
(2)
F vdW共d兲 ⬀
A Hamaker
L m⫹1
(3)
Vol. 83, No. 9
present between the two bodies, the simple assumption of additivity may break down.
Let us now consider in detail the formalism developed by
Hamaker for nonretarded vdW interactions between materials.
(The behavior of retarded vdW interactions is discussed in detail in
Section II(2)(E).) Consider the configuration for two plane-parallel
materials (numbered 1 and 3) shown in Fig. 1(a) that are separated
a distance L by an interlayer (numbered 2) that may be either a
vacuum or another material. The Hamaker constant, ANR, represents the magnitude of the LD interactions and is defined by Eq.
(4) for the interaction energy and by Eq. (5) for the force between
two materials. ANR is defined by convention as being positive for
an attractive force corresponding to a negative LD interaction
energy. ANR is only a function of the material properties of the two
particles and the interlayer material, assuming a uniform interlayer
film.
NR
A 123
⫽ ⫺12␲L 2 E vdW
A
NR
123
⫽ ⫺6␲L F vdW
3
(4)
(5)
This LD interaction is considered long range, because, across a
uniform gap between plane-parallel materials, the interaction
energy varies as a simple power law with L⫺2, and the force varies
with L⫺3, where L is the gap thickness. Most other forces found in
physical systems may be of much larger magnitude, but decrease
much more rapidly with distance.
The three-layer configurations shown in Fig. 1(a) can apply to
many materials systems of interest in ceramics. For example, if
there is a vacuum or an intergranular glassy film between two
particles of the same material—i.e., material 1 and 3 are identical—then we have a symmetrical configuration denoted as 121. If,
instead, the two particles are dissimilar—i.e., material 1 is separated from material 3 by an interlayer of material 2—then we have
an asymmetrical configuration denoted as 123. An additional 123
configuration is the case of a film wetting a free surface, where
material 3 would be vacuum or air.
Ninham and Parsegian42 solved the LD interaction for the
more-complex five-layer configuration, shown in Fig. 1(b), which
is appropriate to a soap-bubble film. In this symmetrical case,
denoted as 12321, material 1 is air, while the three-layer interlayer
is the soap bubble membrane and consists of a surfactant layer of
material 2, which coats the liquid-water layer, denoted as material
3. For the soap film, it is assumed that the surfactant layer has a
fixed thickness, and when the thickness of the film is varied, only
the thickness a of material 3 changes, leading to a scaling for this
ANR
12321 Hamaker constant. If the real variation in the system was
such that material 2 and material 3 would vary in thickness,
thereby keeping the relative quantities of materials 2 and 3 in the
interlayer constant, then the ANR
12321 Hamaker constant is considered
to be on an L-scaling basis and can be completely different from
(B) Continuum Approach for Bulk Interactions: Once the
nature of the individual dipolar interactions was identified, Hamaker12 was able to determine the macroscopic London–vdW
interaction between two spherical particles by summing all dipolar
interactions of the atoms and molecules of a solid or liquid. De
Boer41 calculated the vdW interaction for parallel plane solids,
also by summation of dipolar interactions. This method is based on
the assumption of pairwise additivity of the LD interaction, which
is always met by two bodies under vacuum with no intervening
phase between them (as considered by Hamaker). As presented in
Section II(2), when there is an intervening or interlayer material
Table I. London Dispersion Power Laws for Various
Configurations
Configuration
m
Energy
Force
Molecules: point to point
Two-plane parallel bodies
Two spherical particles
6
2
1
6
2
1
7
3
2
Fig. 1. Schematic geometry for Hamaker constant calculations. (a)
Corresponds to asymmetrical 1–2-3 Hamaker constants, where, if materials
1 and 3 are the same, then this is the simpler case of a symmetrical 1–2-1
Hamaker constant. (b) Represents the more-complex symmetrical fivelayer case, where a layer of material 2 forms on the surface of material 1
while material 3 is the central phase in the interfacial film. (Reprinted with
permission from Elsevier Science.23)
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Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
the a-scaling Hamaker constant. The details of the scaling changes
in the ANR
12321 Hamaker constants are discussed in the paper by
French et al.23 Parsegian and Ninham43 also solved the more
general problem for films with more layers than the 12321 case,
and this is discussed in more detail in Section III(7)(D).
(C) Zero Frequency, IR, and UV Terms in LD Interactions:
The vdW force has three components: the Keesom, or orientation
force; the Debye, or induction force; and the LD force. We can
partition these contributions to the Hamaker constant of the vdW
force, as shown in Eq. (6), so as to evaluate the magnitudes of the
differing dipolar contributions.
A vdW ⬇ A Keesom ⫹ A Debye ⫹ A LD
(6)
The magnitude of the LD force is given by the Hamaker
constant. The Hamaker constant of the LD interaction has three
components, shown in Eq. (7): a zero frequency term, a term that
results from the vibrational contribution observed in the IR part of
the frequency spectrum, and the electronic contribution observed
in the UV part of the frequency spectrum.
A LD ⬇ A zero frequency ⫹ A IR (vibrational) ⫹ A UV (electronic)
(7)
Bergström30 has estimated the magnitude of the IR contribution
to the Hamaker constant. The IR contributions are important in
systems, such as BaTiO3, that have strong vibrational modes.
Water can also exhibit strong low-frequency excitations and,
therefore, in, for example, biological lipid-water systems, lowfrequency contributions to the nondispersion forces and their
temperature dependence can be prominent.44 In most other ceramic systems, the IR contribution is very small.
(2) Lifshitz Theory and Electronic Structure and Bonding
(A) Casimir Force: In 1948, Casimir16 used the recent
developments in quantum electrodynamics (QED) to calculate the
“attraction of two perfectly conducting plates” separated by
vacuum. As a step beyond the simple pairwise additive summation
of dipole/dipole interactions, Casimir was the first to use the fact
that the optical properties of the perfectly conducting metallic
plates are a direct measure of the virtual photon production and the
intrinsic polarizability of the metal plates. This coupling of the
fundamental optical properties to a direct calculation of the LD
force between two bodies was a striking advance in the research.
The attractive force he calculated for two metallic plates came to
be known as the Casimir force. The existence of this theoretically
predicted force stood unconfirmed until the recent work of
Lamoreaux,18 who was able to directly measure it.
(B) Lifshitz Theory: In 1956, Lifshitz19 produced a general
theory of the LD interaction for two solid materials separated by
vacuum, basing the results on QED and quantum field theory.
Lifshitz brought the connection between the fluctuating electric
dipoles and the dielectric properties and electronic bonding of
materials to the forefront. The derivation was formulated directly
in terms of the dielectric and optical properties of the media
involved. For two materials separated by vacuum, Lifshitz found
the interaction obeys pairwise additivity and is always attractive.
(C) Repulsive Forces and the Assumption of Pairwise Additivity: In 1960, Dzyaloshinskii, Lifshitz, and Pitaevskii20 published another paper on two materials separated by a liquid film.
The interlayer film, because of its own dielectric and optical
properties, is also involved in the LD interaction, and it demonstrates the need for the many-body nature of the dispersion
interaction to be considered. When an interlayer material is
present, it effects the virtual photon exchange and the electromagnetic waves that induce the dipolar attraction in the two materials;
the optical properties of the interlayer material 2 must be considered on an equal basis with the optical properties of materials 1 and
3. What we find is that the optical properties of the interlayer
material (material 2) serve to renormalize the optical properties of
the two ceramic grains or particles (materials 1 and 3). The original
assumption of Hamaker, that the fundamental LD interaction is
2121
only pairwise in its nature, is incorrect; the interaction is fundamentally a many-body interaction.
The many-body nature of the LD interaction can radically
change the dispersion force. Let us consider grains of Al2O3 with
interlayers or coatings of SiO2 or air. We use the index of
refraction, n, as a simple measure of the strength of the dispersion;
Al2O3, SiO2, and air have n values of 1.77, 1.5, and 1.0,
respectively. Consider the case shown in Fig. 2(a), where two
grains of Al2O3 are separated by air. The Hamaker constant for this
configuration is 165 zeptojoules (1 zJ ⫽ 10⫺21 J), corresponding
to an attractive intergranular force. Because the air interlayer can
be considered equivalent to a vacuum interlayer, this LD interaction is pairwise in its nature, where only the bonds in the two
grains need to be considered. However, if we change the air
interlayer to a SiO2 interlayer, as shown in Fig. 2(b), then the
Hamaker constant decreases to a value of 26.2 zJ, a dramatic
reduction in the LD interaction. With the SiO2 interlayer, the
interaction is no longer pairwise, and the interatomic bonds in the
SiO2 serve to moderate the interaction of the bonds (or induced
dipoles) in the two Al2O3 grains. The n value of SiO2 produces a
change in the wavelength of the virtual photons propagating
between the Al2O3 grains. In this situation, the virtual photons that
mediate the interaction actually have a longer optical path length
as they travel between the Al2O3 grains. The higher value of n of
the interlayer material in Fig. 2(b) compared with Fig. 2(a)
effectively renormalizes the separation between the grains, such
that the Al2O3 grains can be considered to be at a greater physical
separation, and the attractive dispersion force between the grains is
reduced. The Hamaker constant is much decreased for the SiO2
interlayer.
If the two grains are of dissimilar materials, then the nonpairwise additivity of the LD interaction can change the forces from
being only attractive to being repulsive. This is shown in Fig. 3(a),
where an Al2O3 grain is separated by a SiO2 interlayer from a
MgF2 grain (these materials have n ⫽ 1.77, 1.5, and 1.39,
respectively). When n decreases from material 1 to material 2 to
material 3, then the Hamaker constant is negative, –5.1 zJ in this
case, and corresponds to a repulsive LD force between the Al2O3
and the MgF2 grains. If we change the MgF2 to air, so that the
configuration, as shown in Fig. 3(b), corresponds to an Al2O3 grain
with a surface layer of SiO2, then we have the condition where the
n decreases from Al2O3 (n ⫽ 1.77) to SiO2 (n ⫽ 1.5) to air (n ⫽
1.0), the Hamaker constant is – 40.4 zJ, and the LD force is again
repulsive. To understand this repulsive force, consider that it
causes the air–SiO2 interface to be repelled from the Al2O3–SiO2
interface; the two interfaces of the SiO2 interlayer are repelled
from each other. Thermodynamically, this corresponds to a wetting condition, where the dispersion interaction causing this
repulsive force would produce a thickening of the SiO2 interlayer;
were equilibrium achieved, the SiO2 interlayer would be of infinite
thickness.
(D) LD and Interatomic Bonds: In 1961, Lifshitz produced a
general theory for the LD forces.21,45,46 This formulation shows
Fig. 2. Two configurations and their corresponding attractive LD forces:
(a) two grains of Al2O3 separated by air, ANR
121 ⫽ 165 zJ, and (b) two grains
of Al2O3 separated by SiO2, ANR
121 ⫽ 26.2 zJ.
2122
Journal of the American Ceramic Society—French
Vol. 83, No. 9
of retardation, of which the nonretarded case is one limit. Calculation of retarded dispersion forces is discussed in Section II(5)
and applications are discussed in Section III(8).
Two limiting cases of the Hamaker constant are generally
considered: nonretarded and retarded. When retardation of the
dispersion interaction dominates, then, to first approximation, the
power-law dependence of the dispersion interaction decreases, as
given in Eqs. (9) and (10) for point-to-point interaction. This
corresponds to a decrease in the dispersion force at large separation. Casimir and Polder,47 Lifshitz, and, more recently, Wennerström et al.48 have discussed the origins and effects of retardation,
and Overbeek and co-workers49 in the 1950s measured these
forces, because they could not bring two solids to close enough
approach to avoid the effects of retardation.
Fig. 3. Two configurations and their corresponding repulsive LD forces:
(a) grain of Al2O3 and grain of MgF2 separated by an interlayer of SiO2
glass, ANR
123 ⫽ –5.1 zJ, and (b) Al2O3 grain coated by a layer of SiO2 glass,
ANR
123 ⫽ – 40.4 zJ.
nonret
E disp
⬀
ret
E disp
⬀
that the most effective induced dipoles have frequencies in the
visible through vacuum ultraviolet (VUV) spectral ranges. These
induced dipoles can be interpreted in terms of the underlying
electronic structure of the materials, as resulting from the interatomic bonds. To understand the electronic basis of the LD forces,
consider the schematic model of two grains separated by a vacuum
interlayer, as shown in Fig. 4, with s and p electronic orbitals in
each material. Each of these bonds has a characteristic optical
transition energy Ei that corresponds to a characteristic frequency
␻i and wavelength ␭i, as given by Eq. (8), where ប ⫽ h/2␲ is
Planck’s constant. For the induced dipole interactions to develop,
the characteristic frequency of the dipoles in each grain must be
comparable. Therefore, only the bonds at equivalent frequencies in
each material interact, and the LD force results from the summation of the contributions of each interatomic bond that can find a
corresponding bond of comparable frequency in the adjacent grain.
Because of the requirement for comparable frequencies and
energies for a dispersion interaction to develop, we see that the LD
interaction is inherently based on spectral properties of the
interacting materials and that these spectral properties correspond
to the optical properties of the materials.
E ⫽ ប␻ ⫽
hc
␭
(8)
(E) Retardation of the LD Interaction at Long Distances:
Lifshitz also introduced the concept that retardation of the LD
interaction occurs when the interparticle spacing L becomes large
in comparison to the transit time for the virtual photon across the
interparticle gap. The Lifshitz theory includes the complete effects
Fig. 4. Schematic model of two plane-parallel materials interacting
across a gap. Three electronic interatomic bonds are shown with differing
transition energies and frequencies in each grain. Interactions that develop
between bonds of comparable frequencies contribute to the LD interaction.
1
L6
1
L7
(9)
(10)
Retardation effects on the LD force do more than produce an
accelerated decrease in the power law, as suggested by Eq. (10).
The details of the electromagnetic interaction and the electronic
structure of particular materials can produce new phenomena that
are beyond the predictions of dispersion theory in the nonretarded
regime. For example, dispersion forces can, without the effects of
other interactions, produce surficial films that exhibit equilibrium
thicknesses. This occurs in the case of the surface melting of
ice,50,51 whereby ice exhibits an equilibrium thickness film of
liquid-water on its surface at temperatures below the freezing
point, which results directly from the electronic structure and
interatomic bonding of ice and liquid-water. This surface melting
also is critical in ice skating, frost heaves, and lightning.52
(F) Interband Transitions of the Electronic Structure and
Optical Properties: (a) Electronic Structure: The electronic
structure of materials represents the interatomic ionic, covalent,
and metallic bonds formed by electrons that form the solid.53,54
The electronic structure can be considered from two general
perspectives: that which provides information on the density of
states (DOS) of the valence and conduction bands and that which
supplies information on the joint density of states (JDOS) for
electronic transitions from the occupied valence bands to the
unoccupied conduction bands.55 Experimental techniques that
produce information on the DOS are X-ray and UV photoelectron
spectroscopy (XPS and UPS), inverse photoemission spectroscopy, and energy-loss near-edge structure (ELNES). The DOS also
can be determined from first principles, using band structure
techniques typically under the local density approximation, where
the complete band diagram of the occupied and unoccupied
electronic states is produced, and the density of these states per
unit energy interval is determined.56,57
The LD interaction is an electromagnetic interaction, governed
by Maxwell’s equations, and, therefore, the JDOS perspective is
the essential approach that couples the electronic structure and the
dispersion forces. Consider the optical, interband transitions in the
electronic structure, where the occupied valence band states are the
initial states, and the unoccupied conduction band states are the
final states. The JDOS corresponds to the matrix element for
optical transitions between the valence and conduction band states.
The matrix element is large for allowed transitions between bands
with a large DOS at the transition energy and is smaller if the
relative DOS of the two bands is less. The JDOS is zero if the
transitions are not allowed or if no initial or final states are present
at the transition energy. Because the JDOS includes the optical
matrix element, it corresponds to the optical properties of the
material and to optical transitions interband, i.e., from occupied to
unoccupied states. The optical properties can be determined from
first-principles band structure calculations, if, after completing the
band structure calculation, the optical matrix elements are then
calculated for all possible interband transitions, and these are then
summed so as to produce the JDOS or a related optical property,
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
such as the imaginary part of the dielectric constant, ε2. The optical
properties also can be determined from analysis of experimental
optical data, such as reflectivity or energy loss. This is discussed in
Section II(2)(F)(b).
It is the connection between the electronic structure of the
materials and their optical properties that gives us both components of the LD interaction. The interatomic bonds in, e.g.,
material 1, as represented by the material’s optical properties,
produce the electromagnetic fluctuations that launch the virtual
photons. These photons then interact with the interlayer materials
(material 2) and the other grains (material 3) through Maxwell’s
equations and the optical properties of these other materials.
(b) Optical Properties: The propagation of electromagnetic
waves in materials is governed by Maxwell’s equations, and the
optical properties of solids are the fundamental materials input.58
These optical properties are complex quantities in which the real
and imaginary parts correspond to absorption and dispersion of
light in the material. The dispersion of light is the change of the
wavelength and speed of propagating light under the influence of
the real part of the index of refraction of a material.14 There are
algebraic relations among the numerous optical properties, such as
the index of refraction (n̂ ⫽ n ⫹ ik), dielectric constant (ε̂ ⫽ ε1 ⫹
iε2), optical conductivity (␴ˆ ⫽ ␴1 ⫹ i␴2), and interband transition
strength (Ĵ ⫽ Jcv,1 ⫹ iJcv,2). Moreover, the Kramers–Kronig (KK)
dispersion relations govern the relationship of the real and imaginary components of any one optical property. Here we use the
form presented by Wooten58 for optical property relations, and
NIST59 for fundamental physical constants.
Using VUV spectroscopy to measure optical reflectivity over a
wide energy range, we can determine the optical properties of
materials. (For more information on the experimental details of
VUV and optical reflectivity and spectroscopic ellipsometry, see
French and co-workers.60 – 65 The KK dispersion relations66 – 68
permit the analysis of reflectance, or other optical data, to obtain
any conjugate pair of complex optical properties. The experimental
data must span an infinite energy range and, because, in practice,
experimental data are of limited range, these must be extended by
extrapolation. For the optical reflectance, R(E), and the reflected
amplitude, ␳(E) ⫽ R(E)1/2, the KK transform for reflectance69 (Eq.
(11)) recovers the phase ␪(E) of the reflected wave.
␪ 共E兲 ⫽ ⫺
2E
P
␲
冕
⬁
0
ln ␳共E⬘兲
dE⬘
E⬘ 2 ⫺ E 2
(11)
From the reflected amplitude and phase, any conjugate pair of
optical properties can be computed. The complex index of refraction, for example, is obtained by solving Eq. (12), and the
dielectric function is obtained by Eq. (13).
n ⫺ 1 ⫹ ik
⫽ ␳共E兲e i␪共E兲
n ⫹ 1 ⫹ ik
(12)
ε 1 ⫹ iε 2 ⫽ 共n ⫹ ik兲 2
(13)
2123
Fig. 5. Interband transition strengths, Jcv, of (red) AlN and Al2O3 (blue)
determined from VUV reflectivity.
interband transition strength, neff(E), is
n eff共E兲 ⫽
4v f
mo
冕
E
J cv共E⬘兲
dE⬘
E⬘
(15)
0
where vf is the volume of the formula unit (20.86 Å3 for AlN and
4.25 Å3 for Al2O3) gives the number of electrons contributing to
the optical transitions up to an energy E. The results for AlN and
Al2O3 are shown in Fig. 6.
Electron energy-loss spectroscopy (EELS) is a second experimental method that determines the energy-loss function (ELF ⫽
–Im(1/ε)), another of the optical properties of materials. EELS
spectroscopy can be performed as a bulk or surface measurement
in dedicated EELS spectrometers, but the spatially resolved (SR)
EELS that is now common in electron microscopes has particular
utility in the study of the LD forces.72 SR-EELS can be used to
measure the ELF in the core EELS region (energy range from 200
to ⬎1000 eV), used for “fingerprinting” chemical constituents in
materials; in the ELNES region (energy range from 70 to 500 eV),
used for the conduction band DOS; and in the valence EELS
(VEELS) region (energy range from 0 to 50 eV), used for the
valence to conduction band transitions of the interband electronic
structure. VEELS combined with the high spatial resolution
available in the electron microscope (e.g., a scanning transmission
electron microscope typically has a 0.5 nm probe diameter) is an
exceptional probe of the electronic structure of interlayer materials
and grains in ceramic systems.
We typically render the optical response in terms of the interband
transition strength, Jcv(E), which is related to ε(␻) by Eq. (14),
J cv共E兲 ⫽
m 2o E 2
共ε 共E兲 ⫹ iε 1 共E兲兲
e 2 ប 2 8␲ 2 2
(14)
where Jcv(E) is proportional to the interband transition probability
and has units of g䡠cm⫺3. For computational convenience, we take
the prefactor m2oe–2ប–2 in Eq. (14), whose value in cgs units is
8.289 ⫻ 10⫺6 g䡠cm⫺3䡠eV⫺2, as unity. Therefore, the units of the
Jcv(E) spectra are eV2. The interband transition strength of Al2O3
and AlN, as determined from VUV spectroscopy, are shown in
Fig. 5. The differing energies and transition frequencies of the
interatomic bonds in these two materials are visible in these
spectra.
The oscillator strength sum rule,70,71 given in Eq. (15), for the
Fig. 6. Oscillator strength sum rule per formula unit of (red) AlN and
(blue) Al2O3 determined from VUV reflectivity, showing the number of
electrons involved in the optical transitions.
2124
Journal of the American Ceramic Society—French
The methods of VEELS have been discussed in detail.24,73,74
The experimentally determined ELF is first corrected for multiple
scattering effects using Fourier logarithm deconvolution, then the
ELF is scaled through application of the index sum rule shown in
Eq. (16), where P represents the Cauchy principal part of the
integral.
1
2
1⫺ 2⫽ P
n
␲
冕 冉 冊
⬁
⫺1
Im
ε共E⬘兲
1
d(E⬘)
E
kT
E⫽
2␲
冘冕
⬁
⬁
⬘
n⫽0
␳ d␳ ln G共␰ n 兲
(19)
0
0
冉 冊
1
2
⫽1⫺ P
ε共E兲
␲
冕 冉 冊
⬁
Im
0
⫺1
ε共E⬘兲
E⬘
d共E⬘兲
E⬘ 2 ⫺ E 2
(17)
冕
⬁
0
␻ε 2 共␻兲
d␻
␻2 ⫹ ␰2
␰n ⫽
2␲kT
n
ប
(20)
At this point, we remark on the formalism of the complex
frequency ␻ˆ of the oscillating dipoles defined in Eq. (21) (where
i ⫽ (⫺1)1/2):
␻ˆ ⫽ ␻ ⫹ i␰
From these results, any of the complex optical properties can be
determined. The interband transition strength up to energies of 120
eV for a variety of ceramic materials is shown in Figs. 7 and 8.74
These spectra emphasize the very large differences in the interband
electronic structure and optical properties of materials.
(G) LD Spectrum ε2(␰): The major work involved in calculations of Hamaker constants lies in determining or approximating
the LD spectra ε2(␰) for the materials of interest. Once the LD
spectra have been determined, the particular values of the retarded
or nonretarded Hamaker constant A for a particular configuration
of grains and interlayers constants is determined by computing
integrals of the spectral difference functions G, which are discussed below.
When direct experimental data or first-principles calculations75
on the interband optical properties of the desired materials are
available, we can use the KK relation given in Eq. (18) to
determine the LD spectra following the method of Hough and
White76. The LD spectrum, ε2(␰), is an integral transform of the
imaginary part of the dielectric constant from a function of the real
frequency ␻ to a function of the imaginary frequency ␰.† The LD
spectrum is an optical property and represents the retardation of
the oscillators.
2
ε 2 共␰兲 ⫽ 1 ⫹
␲
can be calculated. Following Lifshitz19 and Ninham and Parsegian,42 the interaction free energy per unit area resulting from the
LD interaction can be given for one or more parallel films
intervening between two media within a planar gap (see Fig. 1(a))
by Eq. (19), where ␰ is defined in Eq. (20).
(16)
Once a single-scattering ELF with a quantitative amplitude has
been determined, then the KK relation shown in Eq. (17) is used to
determine Re(1/ε) from the experimentally determined –Im(1/ε).
Re
Vol. 83, No. 9
(18)
(H) Hamaker Constant ANR: Once the LD spectra have been
determined, then the Hamaker constant for different configurations
For the case of a simple oscillator or sums thereof, LD transform of ε2(␻) to ε2(␰)
can be accomplished by the simple variable substitution of i␰ for ␻ in the oscillator
equation.
†
Fig. 7. Interband transition strengths, Jcv, of three phases of Al2O3
determined from V-EELS with comparison to VUV results for ␣-Al2O3.
(21)
From the interaction energy, we can, using Eq. (4), determine the
Hamaker constant ANR.
冘冕
⬁
A NR ⫽ ⫺6kTL 2
⬁
⬘
n⫽0
␳ d␳ ln G共␰ n 兲
(22)
0
Now the configurations shown in Fig. 1 for the LD forces can
be calculated. These are the nonretarded (denoted NR) Hamaker
NR
constants for three-layer geometries with a single film (ANR
121,A123)
and five-layer geometries with a three-layer intervening film
(ANR
12321). These three types of Hamaker constants can be formulated on a common basis by defining three appropriate versions of
the function G(␰) as follows:
NR
G 121
共␰兲 ⫽ 1 ⫺ ⌬ 212 e ⫺2a ␳
(23)
where a is the thickness of the central layer (layer 2 for (ANR
121,
NR
ANR
123), and layer 3 for A12321),
NR
共␰兲 ⫽ 1 ⫺ ⌬ 32 ⌬ 12 e ⫺2a ␳
G 123
(24)
and
NR
共␰兲 ⫽ 1 ⫺
G 12321
共⌬ 32 ⫹ ⌬ 21 e ⫺2b ␳ 兲 2 e ⫺2a ␳
共1 ⫹ ⌬ 32 ⌬ 21 e ⫺2b ␳ 兲 2
(25)
where b is the invariant thickness of the intervening film between
each particle and the central film in the case of ANR
12321) (Fig. 1). ⌬
is the difference of the LD spectra defined in Eq. (26).
⌬ kj ⫽
ε 2, k 共␰兲 ⫺ ε 2, j 共␰兲
ε 2, k 共␰兲 ⫹ ε 2, j 共␰兲
(26)
Fig. 8. Interband transition strengths, Jcv, of MgO, MgAl2O4, and Si3N4
determined from V-EELS with comparison to Si3N4 determined from VUV
spectroscopy.
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
(3) Full Spectral Nonretarded Hamaker Constants ANR from
Spectral Optical Properties
With spectral optical properties over very wide energy ranges,
as are available from optical reflectivity, VEELS measurements, or
ab initio band structure calculations,75 it is now possible to
calculate Hamaker constants for the LD forces using the Lifshitz
theory. We refer to Hamaker constants determined from this
approach as full spectral (FS) Hamaker constants. The approach is
summarized in Fig. 9. Once FS optical properties have been
determined experimentally or from band structure calculations, the
LD spectra are calculated and stored in a database of materials
properties. These LD spectra are then used to calculate the spectral
difference functions that are appropriate to the configuration of
materials in the calculation of interest.
The details of the calculations of the FS Hamaker constants are
given in Panel A.
(A) Estimated Corrections to FS Hamaker Constants ANR:
(a) Zero Frequency Term in the Lifshitz Equation: The determination of Hamaker constants using the FS method contains the
zero frequency term that results in the integral over frequency.
This zero frequency term is weighted by one-half in the double
summation form of the equations. In this presentation, we have
explicitly shown the double summation form and the 3kT/2
prefactor for clarity. The zero frequency term is correctly30
accounted for and evaluated in the FS Hamaker constant method.23
(b) Neglect of IR Vibrational Results in LD Spectra: In the
calculation of the LD spectra, we use experimental optical properties that typically extend from a low energy of 2– 4 eV to a high
energy of 40 –140 eV. In this way we contain all the interband
electronic transitions in the experimental data range. In materials
that have large dielectric polarizabilities from IR vibrational
modes, these modes produce an IR contribution to the optical
properties that adds to the value of the Hamaker constant. Because
of the form of the integrals, they are heavily weighted to the
electronic contributions resulting in the UV and VUV optical
ranges. However, in systems with very low optical contrast
between the materials—i.e., very-well-matched optical properties
or indexes of refraction for the grains and the interlayer—the
Hamaker constant resulting from the UV-electronic contributions
can be very small, e.g., ⬍1–5 zJ. When there is very low contrast,
even a small contribution to the Hamaker constant from IR
vibrations excitation may be decisive if the contribution is large
enough. Bergström30 discussed this IR vibrational contribution,
evaluating its magnitude for two cases: BaTiO3, which has a very
high zero frequency dielectric constant resulting from IR vibrational modes, and SiO2, a material whose zero frequency dielectric
constant is determined by its UV electronic transitions. Bergström
found that, for BaTiO3, the IR vibrational contribution to the
Hamaker constant is substantial, corresponding to an increase of
the Hamaker constant by as much as 50 zJ for interlayers from
vacuum to hydrocarbon solvents. For a material such as SiO2, the
IR vibrational contribution is negligible, corresponding to an
increase of 1.5 zJ for ANR [SiO2兩vacuum兩SiO2] to a value of 65
2125
zJ. In SiO2, the IR vibrational contribution is important only in the
case where the UV electronic contribution to the Hamaker constant
approaches 0 zJ. The SiO2 case is more characteristic for ceramic
materials, where the IR vibrational contributions are negligible.
(c) Changes in the Spectral Optical Properties: The accuracy of the input optical properties and the resulting LD spectra are
critical to the accuracy of the calculated FS Hamaker constants.
Consider the case of Al2O3, in which the optical properties of
Al2O3 published in 1998,62,64 shown in Fig. 10, have improved
accuracy to those published in 199454 because of a more accurate
implementation of the KK boundary conditions of the reflectance
outside the experimental data range. The 1998 optical property
values led to changes in the amplitude of the interband transition
strength at higher energies. These refinements in optical properties
are not unusual; The Handbook of Optical Constants of Solids77
publishes critiques of the literature values of the optical properties
of materials, such as Al2O3,78,79 and these demonstrate the
consensus values of the optical properties or the disagreements.
The very small changes from 1994 to 1998 correspond only to
changes in the index of refraction, determined by the index sum
rule:70 n(Al2O3–1994) ⫽ 1.75 and n(Al2O3–1998) ⫽ 1.77.
Changes in the FS Hamaker constants of Al2O3 calculated using
the 1998 and 1994 versions of the Jcv spectra are shown in Table
II. Examples are given for vacuum, water, and SiO2 interlayers and
for a surficial SiO2 film on Al2O3. With the largest Hamaker
constant (shown for the vacuum interlayer, because it has the
largest index contrast with Al2O3), ANR
121 increases 20 zJ from 145
to 165 zJ with the 1998 spectra, corresponding to a 14% increase.
For lower-contrast interlayers, such as SiO2, ANR
121 increases from
18 to 26 zJ, corresponding to a 45% increase.
The quantitative accuracy of the optical properties used in FS
Hamaker constant calculations contributes the largest uncertainty
in the magnitude of ANR. These uncertainties in the input optical
properties also impact the simple spectral and index approximation
Hamaker constant calculations discussed in Section II(4), because
these methods are all based on fits to the fundamental optical
properties.
The uncertainties in the absolute magnitude of ANR due to the
accuracy of input spectra do not impact comparisons of and trends
Fig. 10. Interband transition strength, Jcv, of Al2O3 published in (red)
199454 and (blue) 199862.
Table II. Variation in Hamaker Constant ANR because of
Changes in Optical Properties Spectra
A [Al2O3兩vacuum兩Al2O3]
ANR[Al2O3兩water兩Al2O3]
ANR[Al2O3兩SiO2兩Al2O3]
ANR[Al2O3兩SiO2兩air]
NR
Fig. 9. Schematic diagram for the determination of full spectral Hamaker
constants from various forms of spectral optical properties determined
from experiment or theory.
Al2O3 spectra 1994
Al2O3 spectra 1998
145 zJ
44.6 zJ
17.6 zJ
⫺33.9 zJ
164.9 zJ
57.0 zJ
26.2 zJ
⫺40.4 zJ
2126
Journal of the American Ceramic Society—French
Panel A:
Vol. 83, No. 9
Full Spectral Hamaker Constants ANR
(1) Calculation of ␧2(␰) Spectra
Once the complex optical properties as a function of the real frequency ␻ have been determined, the LD integral transform (Eq.
(18)) must be applied. The LD transform requires data over an infinite frequency or energy range, and, therefore, we use analytical
extensions or wings to continue the data beyond the experimental data range. We choose power-law wings of the form Re[Jcv] ⬀ ␻–␣
on the low-energy side of the data and Re[Jcv] ⬀ ␻–␤ on the high-energy side of the data. We have chosen fixed values of ␤ ⫽
3 and ␣ ⫽ 2. The wings are extended to cover from 0 to 100 eV to minimize errors due to neglected areas in the ensuing integrals
used in Hamaker constant determination. On determining the LD spectrum, we retain the complete spectrum over the entire 0 –100
eV range, not the limited experimental data range, to facilitate the evaluation of the spectral difference functions while continuing
to minimize errors resulting from neglected areas between the ε2(␰) spectra.
The interband transition strengths and LD spectra for Si3N4 and an YAl–SiON glass23 are shown in Fig. A1.
Fig. A1. Interband transition strengths and the corresponding LD spectra for Si3N4 and an YAl–SiON glass. Hamaker constant for ANR
[Si3N4兩vacuum兩Si3N4] ⫽ 192 zJ, whereas the Hamaker constant of YAl–SiON glass as the interlayer between the Si3N4 grains, is ANR
[Si3N4兩YAl–SiON兩Si3N4] ⫽ 9.3 zJ. (Reprinted with permission from Elsevier Science.24)
After the LD spectra ε2(␰) are calculated, they are accumulated in a spectral database‡ from which differing combinations can be
used to calculate Hamaker constants. One of the properties of ε2(␰) is that the value at zero energy is the square of the real part
of the index of refraction. Therefore, we can check the validity of the ε2(␰) analysis by comparing the implied index to the known
index of the material in the visible.
(2) Full Spectral Hamaker Constants: Calculation of ANR from London Dispersion Spectra
Once we have determined the complex optical properties and calculated the LD spectra, we can calculate the Hamaker constants
for many physical geometries. However, the integrals given in Eq. (22) of the G functions of Eqs. (23)–(25) are not amenable to
direct evaluation. We need to reformulate the integral over wave vector (d␳) into a summation and to change the infinite frequency
sum into one over the finite data range available experimentally. Here we extend the formulation of Hough and White.76
(A) Integral over the Wave Vector (␳): We simplify the Hamaker constant Eqs. (23)–(25) by first transforming the integral
NR
NR
on the wave vector ␳ into a summation. The results of this transformation are given in Eqs. (27)–(29) for ANR
121, A123, and A12321,
respectively.
1
4a 2
NR
A 121
⫽ 6kTL 2
NR
⫽
A 123
3kT
2
NR
⫽
A 12321
冘冉
⬁
⬘
n⫽0
冘冘
⬁
⬁
⬘
n⫽0
3kT
2
⫹⌬ 212 ⫹
s⫽1
冊
⌬ 412 ⌬ 612
3kT
⫹
⫹... ⫽
8
27
2
冘冘
⬁
⬁
⬘
n⫽0
s⫽1
共⌬ 212 兲 s
s3
(27)
共⌬ 32 ⌬ 12 兲 s
s3
冘 再冘 冋
⬁
⬘
s⫽1
册
⌬ 2s
32
⫹
s3
(28)
冘冋
⬁
s⫽1
册
⌬ 2s
21
⫹
共1 ⫹ 2b/a兲 2 s 3
冘 冋写
⬁
s
s⫽1
k⫽1
册
additional
⌬ s32 ⌬ s21
2共2k ⫺ 1兲
⫹ cross
2 3
k
共1 ⫹ b/a兲 s
terms
冎
(29)
‡
GRAMS is a PC-based spectroscopy environment in which the ELECTRONIC STRUCTURE TOOLS (EST), and HAMAKER.AB, a component of EST, have
been developed. GRAMS is available from Galactic Industries, Salem, NH (http://www.galactic.com). HAMAKER.AB and its spectral database of LD spectra is
available as part of EST from Deconvolution and Entropy Consulting, Ithaca, NY (http://www.deconvolution.com).
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
Panel A:
2127
Continued
The ANR
12321 equation can be used for values of b such that the volume fraction Vf of material 2 in the film, defined as Vf ⫽ 2b/L,
varies from 0 ⬍ Vf ⬍⬍ 1, subject to the approximations that permit the expansions, e.g., ⌬22⌬21e–2b␳ ⬍⬍ 1.
(B) Summation over Frequency (d␰): These integrals have been written for evaluation over an infinite imaginary frequency
range in ␰; however, we have determined the LD spectra for a range of 0 to 100 –150 eV. Moreover, the LD spectra have a
sampling interval ␦␰ that equals the experimental data-point spacing. Therefore, to cover the general case, where many spectra are
of variable energy range and sampling density, we first select an upper energy limit for summation, Elim, which is determined by
the smallest energy range available, and then we resample the LD spectra at 0.054 eV (T ⫽ 100 K). With this sampling interval,
we are in a position to calculate Hamaker constants at any temperatures that are multiples of 100 K. Therefore, the equations used
herein to determine the Hamaker constants for the various cases become, in the explicit form of a double summation, as follows:
NR
A 121
⫽
NR
A 123
⫽
3kT
2
3kT
2
冘冘
共⌬ 212 兲 s
s3
(30)
冘冘
共⌬ 32 ⌬ 12 兲 s
s3
(31)
E lim
4
⬘
n⫽0
s⫽1
E lim
4
⬘
E⫽0
3␦E
NR
A 12321
⫽
4␲
s⫽1
冘 冦冘 冋
E lim
4
⬘
E⫽0
s⫽1
册 冘冋
⌬ 2s
32
⫹
s3
4
s⫽1
册 冘 冋写
⌬ 2s
21
⫹
共1 ⫹ 2b/a兲 2 s 3
4
s
s⫽1
k⫽1
册
12
⌬ s32 ⌬ s21
2共2k ⫺ 1兲
additional
⫹ cross
k
共1 ⫹ b/a兲 2 s 3
terms
冧
(32)
In these equations, the primed summation represents that only one-half of the first term in the frequency (energy) sum is
counted, and the summation is understood to include only terms that satisfy the sampling interval ␦E ⫽ 2␲kT. The
temperature-dependent sampling interval results from the presence of coth(ប␻/2kT) in Lifshitz’s19 molecular force equation (Eq.
(24)), which changes an integral over all frequencies to a sum. The one-half weight given to the zero energy term was a necessary
step when the integral over all frequencies was changed to an integral over positive frequencies.
in Hamaker constants for a series of materials configurations. The
variation of ANR with differing interlayers shown in Table II is
comparable for both of the Al2O3 spectra used.
(d) Neglect of (d␰) Frequency Integral from 100 eV to Infinity: In the FS Hamaker constant method, we typically restrict
the range of the LD spectra to an energy of 100 eV, whereas the
integrals over frequency should extend to infinity. We neglect the
contribution to the integrals or sums of the spectral difference
functions, ⌬kj, given by Eq. (26), from 100 eV to infinity. We can
estimate the “missing” contribution to the integrals of the LD
spectra from 100 eV to infinity by recognizing that, at high
energies, ⌬kj is proportional to E–2. At high energies, the summations on s in Eqs. (30)–(32) are proportional to E⫺4 plus other
terms that decrease more rapidly with increasing energy. Therefore, to estimate this high-energy (⬎100 eV) contribution to ANR,
we fit an E⫺4 wing to the energy spectrum and then analytically
integrate this term. We calculate the high-energy contribution
NR
for ANR
121- and A123-type Hamaker constants with high and low
optical contrast between the grains and the interlayer materials,
and we summarize these in Table III. These high-energy corrections to the Hamaker constant are ⬍1 zJ in all cases and
correspond to errors on the order of 0.1%– 0.8%. This estimate of
the high-energy contribution is usually larger than the temperature
dependence of the Hamaker constant from 0 to 300 K because of
the kT terms and sampling effect, which is discussed in Section
II(3)(C)(e).
(e) Temperature Dependence of Hamaker Constant ANR: In
the calculation of the Hamaker constant, the summation over
frequency shown in Eqs. (30)–(32) requires a temperaturedependent sampling of the ⌬kj in units of ␦E ⫽ 2␲kT, along with
the explicit kT prefactor. These two effects produce an explicit
temperature dependence of the Hamaker constant. The temperature
dependence due to the kT prefactor and frequency sampling for the
NR
ANR
121- and A123-type Hamaker constants is presented in Table IV.
Table IV shows that this temperature dependence is, in all cases,
⬍0.2 zJ and corresponds to a contribution to the Hamaker constant
of ⬍0.3% for a temperature range of 2000 K.
In addition to the temperature dependence explicit in the kT
prefactor and the frequency sampling, a much larger contribution
to the Hamaker constant results from the temperature dependence
of the electronic structure and optical properties of materials. It has
been shown53 that the band gap of ␣-Al2O3 decreases with
increasing temperature at the rate of 1.1 meV/K. The index of
refraction of Al2O3 as a function of temperature is shown in Fig.
11. Initially, the index increases, only to decrease for temperature
above ⬃1000 K. The temperature dependence of the ANR
1v1 Hamaker constant shows a similar increase and subsequent decrease
at higher temperatures. This type of second-order variation in the
Hamaker constant, resulting from the experimentally determined
changes in the interband electronic structure, goes beyond any
simple theories of the temperature dependence of the Hamaker
constant and LD force. Moreover, the changes in the Hamaker
Table III. Hamaker Constant ANR and the Contribution to ANR because of Frequency Summation from 100 eV to Infinity
Material 1
NR
A [material
ANR[material
ANR[material
ANR[material
1兩vacuum兩material 1]
1兩water兩material 1]
1兩SiO2兩material 1]
1兩SiO2兩air]
TiO2
Al2O3
SiO2
149.2 ⫹ 0.15 zJ
55.95 ⫹ 0.01 zJ
29.12 ⫹ 0.002 zJ
⫺36.14 ⫹ 0.02 zJ
164.9 ⫹ 0.82 zJ
60 ⫹ 0.39 zJ
26.16 ⫹ 0.22 zJ
⫺40.4 ⫺ 0.21
68.2 ⫹ 0.19 zJ
6.628 ⫹ 0.024 zJ
2128
Journal of the American Ceramic Society—French
Table IV. Temperature Dependence of the Hamaker
Constant because the kT Prefactor and Frequency Sampling
Effects
0K
300 K
2000 K
ANR[Al2O3兩vacuum兩Al2O3] 164.85 zJ 164.851 zJ 164.864 zJ
ANR[Al2O3兩water兩Al2O3]
56.998 zJ 56.9786 zJ 56.8087 zJ
ANR[Al2O3兩SiO2兩Al2O3]
26.1565 zJ 26.1568 zJ 26.1604 zJ
NR
A [Al2O3兩SiO2兩air]
⫺40.4044 zJ⫺40.4047 zJ ⫺40.4081 zJ
Panel B:
Vol. 83, No. 9
Index Approximations of ANR
(1) Tabor Winterton Approximation
Assuming the optical properties of the necessary materials can be described as resulting from a single LO and using
some additional approximations, an analytic approximation
93
for ANR
can be obtained as94
123, the TWA
TWA
A 123
⫽
3␲ប␯ e
4 冑2
2
2
2
2
关共n vis0,
1 ⫺ n vis0, 2兲共n vis0, 3 ⫺ n vis0, 2兲兴
2
2
2
2
1/ 2
1/ 2
共n vis0,
/兵共n vis0,
1 ⫹ n vis0, 2兲
3 ⫹ n vis0, 2兲
2
2
2
2
⫻ 关共nvis0,1
⫹ nvis0,2
兲1/ 2 ⫹ 共nvis0,3
⫹ nvis0,2
兲1/ 2兴其
(36)
which depends on the limiting values of the index of
refraction in the visible (nvis), extrapolated to zero energy
(nvis0), for the i materials (nvis0,i), and the characteristic
absorption frequency (␯e), which is related to the bandgap
and is assumed to be equivalent for all three materials so as
to permit the integrations. When materials 1 and 3 are
identical, the TWA reduces to
TWA
⫽
A 121
2
2
2
3␲ប␯ e 共n vis0,
1 ⫺ n vis0, 2兲
2
2
3/ 2
8 冑2 共n vis0, 1 ⫹ n vis0, 2兲
(37)
The TWA is predicated on the assumption that the predominant contribution to the LD forces results from interband
transitions in or near the visible and UV range associated
with the primary bandgap of the material. Typically, a
characteristic absorption frequency of ␯e ⫽ 3 ⫻ 1015 Hz is
used in TWA calculations
Fig. 11. Temperature dependence of the Hamaker constant ANR
1v1 and the
index of refraction for ␣-Al2O3 up to 1925 K. (Reprinted with permission
23
from Elsevier Science. )
constant are large, varying from 140 to 148 zJ and then decreasing
to 125 zJ, which correspond to variations in the Hamaker constant
of 6%–11%.
(4) Methods for Estimating Hamaker Constant ANR
Where extensive experimental data are not yet available, simple
spectral models or index approximations can be used to estimate
the Hamaker constant for various configurations. Because Lifshitz
theory for calculating A is based on the use of optical properties as
the fundamental input, various methods have been developed to
estimate the optical properties for materials over the wide energy
range needed. These methods25 can be divided into two general
classes: the simple spectral methods,30,76 which use optical data in
the visible to develop parametric Lorentz oscillator58 models of the
LD spectra over a wide energy range,80 and index approximations,
such as the Tabor Winterton approximation (TWA), which are
based on the index of refraction as a measure of a material’s
dispersion.
(A) Simple Spectral Method: To calculate Hamaker constants based on the Lifshitz theory, Ninham and co-workers32,42
introduced the simple spectral (SS) method,30,76,81 which uses a
set of parametric damped Lorentz oscillators to represent the LD
spectrum. Essentially, these oscillators correspond to absorptions
in the material at certain frequencies of given strengths, and these
oscillators can be fitted to physical property data, such as absorption spectra, refractive index, or dielectric constant. A model LD
spectra can be synthesized from the relation
(2) Single Oscillator Approximation
The single oscillator approximation (SOA)87 (Eq. (38)) is
another Hamaker constant approximation similar to the
TWA. However, instead of assuming the characteristic
absorption frequency ␯e is constant, the absorption frequency in the SOA varies with the index of refraction such
that, as the index increases, the oscillator energy decreases.
This leads to index-based estimates of the Hamaker constant
that are comparable to FS Hamaker constants.
A 121 ⫽
312共n 21 ⫺ n 22 兲 2 zJ
关共n ⫺ 1兲 ⫹ 共n 22 ⫺ n 21 兲 1/ 2 兴共n 21 ⫹ n 22 兲 3/ 2
2
1
1/ 2
冘
N
ε共i␰兲 ⫽ 1 ⫹
i⫽1
Ci
1⫹
冉冊
␰
␻i
2
(38)
(33)
where Ci ⫽ (2/␲)(fi/␻i) and fi the strength of the oscillator at
frequency ␻i. After various simplifying assumptions and approximations, they arrive at the following expression for the Hamaker
constant:
SSM
⫽
A 121
3kT
2
冘冘
⬁
⬁
⬘
n⫽0
s⫽1
共⌬ 12 兲 2s
s3
(34)
where ⌬ij are as in Eq. (26), but ␰ is replaced with ␰n ⫽ n(2␲kt/ប).
Ci can be calculated from the refractive index in the visible from
a Cauchy plot,76 which gives, e.g., for a UV oscillator,
n 2 共␻兲 ⫺ 1 ⫽ 共n 2 共␻兲 ⫺ 1兲
␻2
2 ⫹ C UV
␻ UV
(35)
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
Hence, with relatively little data, we can calculate approximations
to Hamaker constants using a minimal amount of information on
the material’s electronic structure.
Bergström,30,82,83 among others,84 has used the SS method to
estimate Hamaker constants for materials separated by interlayer
materials of vacuum (air), water, SiO2, and hydrocarbon solvents.
They calculated their Ci oscillator parameters from optical data
obtained using spectroscopic ellipsometry or from the literature
and the Cauchy plot for the UV terms. The IR terms were
calculated from the relation CIR ⫽ ε(0) – CUV – 1. For ceramic
materials, they typically used one IR and one UV term. For water,
they used several oscillators and fitted them to various published
spectral data of water.
(B) SS Representations of Water: The great importance of
the LD forces in aqueous colloidal systems has made the SS
representations of water much discussed in the literature by
Bergström83 and others.85 Pashley86 has compared the Hamaker
constants for aqueous systems determined from the FS and SS
methods and also from direct force measurement determinations.87
These aqueous Hamaker constant calculations are based on literature data for the spectral optical properties of water.88 –92
(C) Index Approximations: For materials in which spectral
optical properties are not available, the TWA and single oscillator
(SOA) index approximations for the Hamaker constant provide
useful estimates based on readily available indexes of refraction.
Moreover, the variations of Hamaker constants with changes in the
index of refraction of the grains or interlayers are very useful
estimates of trends in LD forces.
The accuracy of TWA estimates is shown in a comparison of FS
and TWA Hamaker constants. Figure 12 shows the Hamaker
constants ANR
1v1 for a vacuum gap between two identical materials,
derived from the FS method and TWA. Agreement between FS
and TWA Hamaker constants is within 15% for materials with an
index of refraction in the range of 1.4 –1.8. However, the TWA is
increasingly poor for materials with higher indexes. Materials with
higher indexes of refraction tend to have lower bandgap energies,
which is neglected with the TWA’s use of the constant ␯e
frequency. For higher-index materials, the SOA provides values
for the Hamaker constant that are closer to the FS Hamaker
constants.
(D) Simple Hamaker Constant Relationships: Various approximate relations have been proposed that relate values of
various Hamaker constants, such as given in Eqs. (39)–(42).
2129
NR
NR
A 121
⫽ A 212
(39)
NR
NR
NR 1/ 2
A 123
⬇ ⫾共 A 121
A 323
兲
A
NR
1v3
⬇ 共A
NR
1v1
A
(40)
兲
NR 1/ 2
3v3
(41)
NR
NR
NR
NR
A 121
⬇ A 1v1
⫹ A 2v2
⫺ 2 A 1v2
(42)
These types of relations, including other similar ones given by
Israelachvili,29 can be of use with FS or estimated Hamaker
constants if they are sufficiently accurate.
(5) Retardation of Dispersion Forces and the Retarded
Hamaker Constant AR
The nonretarded Hamaker constant applies when the interparticle separation is very small compared with the wavelength of the
interband absorptions of the interatomic bonds. When this nonretarded condition is met, the interactions among the induced dipoles
can be considered instantaneous, and the photon transit time
between the dipoles can be neglected.
When the interparticle separation is large, e.g., in the range of
5–10 nm, then the transit time of the virtual photons becomes
important. The highest-energy interatomic bonds go out of phase
with each other; therefore, their contribution to the LD interaction
decreases. As the separation increases, the contributions of the
high-energy bonds to the LD force are overestimated in the
nonretarded Hamaker constant, and the retarded interaction (AR)
must be calculated.
To understand the effect of retardation, consider a material with
interatomic bonds at interband transition energies of 7, 14, and 21
eV. The energy of the interband transitions (Eibt) and their
wavelength (␭ibt) are given by Eq. (43) and in Table V. The critical
interlayer thickness at which a particular interatomic bond
dephases is estimated by Eq. (44). For this example, at a film
thickness of 9 nm, the 21 eV transition energy bond no longer
contributes to the dispersion force. When the film thickness
reaches 14 nm, then the 14 eV transition energy bond no longer
contributes. The details of the electronic structure and the interatomic bond energies dictate the details of the impact of retardation on the LD force.
␭ ibt ⫽
hc
E ibt
retard
⬇
L crit
(43)
␭ ibt
2␲
(44)
III.
Applications
The LD forces are fundamental and ubiquitous, spanning all
areas of science and technology. In this section we discuss various
areas where dispersion forces play a role. Because LD forces are
usually one of many forces and effects involved, the magnitude
and effects of the dispersion forces compared with other forces and
phenomena are examined for specific classes of materials. We look
at dispersion forces in colloidal systems, interface energies, and
wetting. Next, intersurface forces with a role in adhesion and
sintering, which result from dispersion interactions, are discussed.
We present the mechanical and electrical properties of a broad
range of ceramic systems that exhibit intergranular glassy films
mindful of the importance these films have acquired in the past 15
years for our understanding of ceramics processing and properties.
Table V. Comparison of Interband Transition Energy and
Wavelengths, and Critical Thickness Estimates for
Retardation Effects in London Dispersion Forces
Fig. 12. ANR
1v1 vs nvis0 for various materials calculated using FS calculations and the TWA. TWA results have been evaluated with a single value
of ␯e of 3 ⫻ 1015 Hz for all materials. (Reprinted with permission from
Elsevier Science.23)
Eibt
␭ibt
retard
Lcrit
Transition energy (eV)
Transition wavelength (nm)
Critical thickness (mm)
21
59
9
14
89
14
7
177
28
2130
Journal of the American Ceramic Society—French
Panel C:
We then finish with a discussion of the effects of retardation on
wetting, which can lead to unique wetting conditions.
Calculation of Full Spectral Retarded
Hamaker Constants AR
We calculate the retarded Hamaker spectrum, i.e., the
Hamaker constant, as a function of thickness, following
Elbaum and Schick.50 The free energy, per unit area, of an
interlayer film of material 2 and thickness L between two
other materials 1 and 2 is given by
E共L兲 ⫽
冋 册冘 冕 再 冋
冋
⬁
kT
8␲L2
⬘
n⫽0
dxx ln 1 ⫺
␶n
⫹ ln 1 ⫺
册
共x ⫺ x1兲共x ⫺ x3兲 ⫺x
e
共x ⫹ x1兲共x ⫹ x3兲
共ε 3 x ⫺ ε 2 x 3 兲共ε 1 x ⫺ ε 2 x 1 兲 ⫺x
e
共ε 3 x ⫹ ε 1 x 3 兲共ε 1 x ⫹ ε 2 x 1 兲
册
(45)
冉
冋
x M ⫽ x 2 ⫺ r 2n 1 ⫺
εM
ε2
冊册
1/ 2
共M ⫽ 1, 3兲
(46)
ε1, ε2, ε3 are the LD spectra of the materials, evaluated at
⬘
imaginary frequencies. The sum ⌺n⫽0
is evaluated at the
sequence of imaginary frequencies ␰n ⫽ (2␲kT/ប)n. Combining this definition of frequency with the constant factor
kT/8␲L2, we arrive at Eq. (47) for free energy of the LD
interaction:
E共L兲 ⫽
冋
册冘
冕 再冋
冋
ប
16␲ 2 L 2
⬘n⫽0 ⌬␰
⬁
⫻
dxx ln 1 ⫺
rn
⫹ ln 1 ⫺
共 x ⫺ x 1 兲共 x ⫺ x 3 兲 ⫺x
e
共 x ⫹ x 1 兲共 x ⫹ x 3 兲
册
共ε 3 x ⫺ ε 2 x 3 兲共ε 1 x ⫺ ε 2 x 1 兲 ⫺x
e
共ε 3 x 3 ⫹ ε 2 x 3 兲共ε 1 x ⫹ ε 2 x 1 兲
Vol. 83, No. 9
册冎
(47)
The sum ⌺⬘n⫽0 over imaginary frequencies ␰n is evaluated at
frequencies (or energies) separated by ⌬␰ ⫽ 2␲kT/ប. When
T ⫽ 300 K, ⌬␰ ⬇ 3.9 ⫻ 103 Hz (0.162 eV). We evaluate the
sum over a finite region from 0 to 100 eV instead of the
infinite sum shown.
Each term of the sum, which is an integral over distance
兰r⬁n dx holding the frequency constant, is also approximated
by a sum in our calculation. The sum extends from the lower
limit of the integral x ⫽ rn ⫽ 2L␰n(ε2)1/2/c to an upper limit
of x ⫽ 10, where the exponential term e–x produces a
negligible integrand. The sampling interval of the sum is
⌬x ⫽ 2000/(10 – rn).
The algorithm for calculation of the retarded Hamaker
constant is as follows:
● Choose a value of L (L ⫽ 0, 1, 2, . . . ) in units of 0.1
nm, where L is the thickness of the sandwich.
● Choose a frequency ␰ and obtain the value of the LD
spectra of the three materials ε1(␰), ε2(␰), ε3(␰).
● Prepare to do an inner integral (integral on x) for this
frequency. Calculate the lower limit of integral, which is
2L␰(ε2(␰))1/2/c. Evaluate the integral. If L ⫽ 0, then multiply the results by one-half.
● Choose a new frequency and repeat the second step
until all frequencies in the LD spectrum are satisfied.
This completes the calculation of the free-energy E(L) for
a single value of the interlayer thickness L. The calculation
is repeated for subsequent interlayer thicknesses L. The
Hamaker spectrum or Hamaker constant as a function of
interlayer thickness is then calculated from the free energy
as a function of thickness using Eq. (2). Finally, the force is
calculated from the free-energy curve, F(L) ⫽ –⌬E(L)/⌬L,
using numerical differences.
(1) Classes of Materials
When assessing the importance of the dispersion forces in a
particular problem, it is essential to consider the class of materials
involved. In hydrocarbon materials and polymers, all the bonding
is covalent and fully satisfied in the molecule or the polymer chain.
These materials have very low levels of unsatisfied bonding, and
their interfacial and surficial properties can be dominated by LD
interactions. A result of the fully satisfied bonding in these
materials is that surface energies tend to be low, and result
predominantly from the LD interaction.
In ceramics and semiconductors, free surfaces typically have
many unsatisfied atomic bonds. This disruption of the bulk
electronic structure corresponds to thermodynamic energies that
are much larger than the contribution resulting from the dispersion
interactions. The surface energies of ceramics are much larger than
polymers, and dispersion contributions, as shown below, are
typically only 10% of the surface energy. In ceramics wet at high
temperature by glasses, the unsatisfied chemical bonds of the
ceramic grains can become satisfied, and then the dispersion
interaction is critical. Silicates and other glass formers have a
polymeric structure, and a high level of their atomic bonding is
satisfied in the structural tetrahedra and the polymer chains.
Silicate surfaces have lower levels of unsatisfied atomic bonds.
In hydrocarbons and hydrocarbon polymers, most chemical
bonding is satisfied and dispersion is important. In aqueous
systems, there are the effects of hydrogen bonding plus dispersion.
In ceramics, ionic bonding is important, and in metallic systems
there is metallic bonding.
(2) Surface Energies, Wetting, and Contact Angles
Because of the essential nature of surface energies and wetting,
there have been many excellent discussions of their role in various
materials systems and applications. Johnson95 focused on wetting
of polymers by hydrocarbon solvents, a situation in which LD
dominates. Israelachvili96 discussed the dispersion force contribution to works of adhesion and contact angles. There even has been
recent discussion of the role of vdW forces in the adhesion of
gecko foot hairs to surfaces.97 Zisman98 and others99 compare
wetting of both low- and high-energy solids, and they focus on the
measurement of the contact angle, because this is a measurement
of surface and interface energy. Clarke has discussed the role of
dispersion and other forces,100 such as electrical double-layer
forces,101 on the wetting of surfaces and interfaces (grain boundaries) in ceramic systems.102
The surface and interface energies of materials are related to
unsatisfied bonding and to the dispersion effects from the optical
contrast. It is these thermodynamic energies that help determine
the wetting of solids by liquids. The topic of surface and interface
energies, and also the contact angle at the triple point between the
solid, liquid, and gas phases, shown in Fig. 13, was first studied by
Young2 and has been an active topic in science ever since. The
Young equation (Eq. (48)) relates the contact angle to the surface
and interface energies of the solid and liquid, through the use of a
force balance at the interface. The closely related Young–Dupre
equation (Eq. (49)) then defines the work of adhesion as the
difference between the free surface energies of the solid and the
liquid and the solid–liquid interface energy.
␥ liquid cos ␪ ⫽ ␥ solid ⫺ ␥ solid⫺liquid
(48)
W ad ⫽ ␥ solid ⫹ ␥ liquid ⫺ ␥ solid⫺liquid
(49)
The thermodynamics of wetting has spanned from the work of
Gibbs6,7 and van der Waals5 to the critical wetting theory of
Cahn103,104 and the development of the effective interface potential approach.105 The study has progressed to the detailed variations of three-dimensional droplets and their contact lines.106 In
the more-complex issue of reactive wetting, e.g., of ceramics by
metals at elevated temperature, the chemical reactions tend to
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
Panel D:
Fig. 13. Schematic drawing of a liquid droplet on a solid substrate (␥sol.
is the solid surface energy, ␥liq. the liquid surface energy, and ␥sol.-liq. the
solid–liquid interface energy).
dominate the wetting process, and the dispersion forces can be
considered negligible. A step beyond the equilibrium wetting
condition is the process by which a wetting film dewets a surface,
i.e., under what conditions a wetting film ruptures and dewets the
substrate.107,108
Polymer– ceramic interactions are important in colloidal processing of ceramics and in engineering polymers, many of which
are ceramic-particulate filled. In the case of milling and extrusion
of ceramic-filled engineering polymers, we consider the dispersion
contributions to the interface energies of polymers wetting SiO2
and Al2O3. Surface energies in most polymers result from the LD
interaction. We can use experimental optical properties to determine their total surface energies. In Fig. 14 we show the interband
transition strengths for five polymers. Polyester, polycarbonate,
and polystyrene contain aromatic or conjugated phenyl groups that
peak in the region from 4 to 7 eV, whereas poly(methyl methacrylate) (PMMA) and poly(ethylene-co-methacrylic acid) are only
linear chain polymers and lack these lower-energy transitions.
From the interband transitions, we can calculate the LD spectra,
NR
and from these the Hamaker constants for ANR
1v1 and A121, where we
choose material 2 as water, SiO2, and Al2O3. The Hamaker
constant ANR [PMMA兩Al2O3兩PMMA] is 24.1 zJ, and, using Eq.
(39), this also corresponds to the case of two grains of Al2O3
separated by a PMMA interlayer. Because all the Hamaker
constants reported in Table VI are symmetrical, i.e., of the ANR
121
type, they are all attractive, and the values are all positive. The
effects of optical contrast are evident for the case of water and
SiO2 interlayers, where the index of the interlay and the polymeric
grains are very similar, the optical contrast in the system is very
low, and the LD interaction and Hamaker constants are small.
On calculating the necessary ANR
1v3 Hamaker constants, we can
determine the dispersion contribution to the interface energies for
the polymer– ceramic systems, as are shown in Table VII. The
2131
Interface Energies and Contact Angles for
Droplet on Substrate
Because the Hamaker constant represents the LD interaction energy of two materials, it gives us the dispersion
contribution to surface and interface energies. If dispersion
forces are the main contribution to the interface energies, we
can also calculate the contact angle. Let us consider a ANR
Hamaker constant for a material, which represents the LD
interaction for two grains of material 1 separated by
vacuum. Let us also assume for now that this is a material,
such as a polymer, in which the surface energy is determined by the dispersion interactions. If we evaluate the
Hamaker constant at a value do, which corresponds to an
equilibrium separation, such as an interatomic bond length,
then we have calculated twice the surface energy of material
1. Typically, the value of do is taken to be 0.165 nm.29
Similarly, if we evaluate the ANR
1v3 Hamaker constant between two different materials 1 and 3 for an appropriate do
separation, again separated by vacuum, then we have
calculated the interface energy for an interface of material 1
and 3. To calculate the surface and interface energies, as
shown in Fig. 13, requires the calculation of ANR
[liquid兩vacuum兩liquid] and ANR [solid兩vacuum兩liquid], then
using Eqs. (50) and (51), resulting in ␥liquid. and ␥solid–liquid.
␥ material1 ⫽ ⫺
A 1v1
1
E
⫽
2 material1 24␲d 2o
␥ material1 material3 ⫽ ⫺
1
A 1v3
E material1 material3 ⫽
2
24␲d 2o
(50)
(51)
These correspond to the LD contribution to the interface
energies. In materials classes where Eqs. (50) and (51)
correspond to the total interface energies, then we can use
the Young equation (Eq. (48)) to determine the contact
angle ␪, which in terms of Hamaker constants and assuming
a constant value of do, is given in Eq. (52).
冉
␪ ⫽ arccos
A 121
⫺1
A 123
冊
(52)
Fig. 14. Interband transition strengths, Jcv, of polyester (PEST), polycarbonate (PCRB), polystyrene (PSTY), poly(methyl methacrylate) (PMMA), and
poly(ethylene-co-methacrylic acid) (PEMA) determined from VUV reflectivity.
2132
Journal of the American Ceramic Society—French
Vol. 83, No. 9
†
Table VI. Refractive Indexes and Hamaker Constants (ANR
121) for Various Polymers
Polymer
refractive index
Polymer
(material 1)
ANR (zJ) (vacuum)
(n ⫽ 1.0)
ANR (zJ) (water)
(n ⫽ 1.33)
ANR (zJ) (silica)
(n ⫽ 1.54)
ANR (zJ) alumina
(n ⫽ 1.76)
1.4893230
1.51231
1.64–1.67
1.70
1.586232
1.590–1.592230
PMMA
ET-MAA
PEST
PIMI
PCARB
PSTY
58.4
47.7
60.9
62.6
50.8
55.6
1.47
2.16
4.05
5.23
3.5
3.16
0.634
2.71
1.94
2.61
3.08
2.04
24.1
33.0
25.4
25.6
32.4
28.3
†
NR
NR
Polymer is defined as material 1, and vacuum, water, SiO2, and Al2O3 are defined as material 2. By definition, A121
⫽ A212
.
Table VII. Dispersion Contributions to Surface and
Interface Energies†
Material
Surface energy
(mJ/m2)
Interface energy on
SiO2 (mJ/m2)
Interface energy on
Al2O3 (mJ/m2)
PMMA
ET-MMA
PEST
PIMI
PCARB
PSTYR
Al2O3
SiO2
29 (41.1)
24
30 (44.6)
30 (41)
25 (33)
27 (40.7)
71
33
62
55
62
63
57
60
89
80
90
90
82
86
†
Assuming do ⫽ 0.165 nm. Literature233 values of the surface energy of polymers
are given in parentheses.
Fig. 15. Force– distance curve for (– – –) LD force, (–䡠䡠–) repulsive force,
and (—) total force.
tabulated surface energies of the polymers are very low, in the
range of 25–30 mJ/m2, and correspond well to independently
determined values of the surface energies.109,110 For Al2O3, the
dispersion force contributions to the surface energies correspond to
only 5% of the measured surface energy, because the unsatisfied
bonding on the surface of Al2O3 contributes a very large thermodynamic energy to the surface energy, dwarfing the contributions
resulting from dispersion.
(3) Balance of Forces
Just as consideration of the material classes involved are
important, other compensating forces usually effect systems as
well as the dispersion interactions. For example, an attractive
dispersion force typically is counterbalanced by a repulsive force,
such as a hard-core repulsion, when two atoms approach. The
outcome is shown in Fig. 15, where the force balance of these two
forces leads to a total force, with an equilibrium point resulting
where the force is zero. This concept of the force balance, shown
in Eq. (53), is common to many applications of dispersion forces,
and the forces involved can result from electrical double layers
(EDL), adsorption (Ads.), hydrogen bonding (HB), and steric
forces (St.), among others.
F ⫽ 0 ⫽ F LD ⫹ F EDL ⫹ F ads ⫹ F HB ⫹ F St
(53)
(4) Colloidal Systems
Colloidal systems of polymer- or ceramic-particulate dispersions serve as critical role in ceramic processing as has been
discussed recently in detail by Lewis.111 The LD forces are a
fundamental attractive force involved in colloidal dispersions,33,112 and it is the detailed balance of this and other forces
that dictates the behavior of the particulate dispersion.22 Derjaguin
and Landau112,113 and Verwey and Overbeek33 (DLVO) developed DLVO theory from the perspective of the force balance of
Eq. (53), which is the basis of understanding and controlling the
interparticle forces and thereby colloidal dispersions stability and
rheology.
(5) Interparticle Forces
The direct measurement of the interparticle force balance has
been a long-standing focus of research, because LD forces and the
force balance can allow stable equilibrium behavior in colloidal
systems. DLVO theory33 has provided a method to estimate the
contributions from the LD force and EDL force, among others, to
interpret measured force– distance relations for two planar bodies,
a sphere or cylinder approaching a plane, two crossed cylinders, or
two spheres. With this theoretical underpinning, work commenced
in the 1950s on developing quantitative tools to directly measure
the interparticle forces. The surface force apparatus (SFA), which
began development in the 1950s, has been joined in the 1980s by
the atomic force microscope (AFM).
(A) Surface Force Apparatus: In the 1950s, Overbeek49,112
and Derjaguin22,114 began measuring interparticle forces and
force– distance relations using plane-parallel plates; they later had
more success using curved surfaces, which simplified alignment of
the two materials. They introduced optical interferometric techniques to accurately measure the separation of the two materials, as
long as they were transparent. These initial results tended to
demonstrate forces much larger than expected for the LD interaction, but with sufficient care, dispersion forces at separations of
100 nm were measured. With the work of Tabor and Winterton93
and Israelachvili,115 the SFA, based on a crossed cylinder geometry, became a well-established instrument for measuring the
intersurface force– distance relations for mica under vacuum or a
wide variety of liquids. Mica is the most common material used in
the SFA because of the ease of preparing an atomically smooth
surface and adhering it to cylindrical substrates. With additional
work at preparing curved surfaces of additional materials, such as
Al2O3,116 the range of materials studied using the SFA expanded.
A large amount of the SFA research was focused on measurement
of the various force terms in DLVO theory for complex liquid
systems.117 Moreover, the effects of the molecular ordering of
organic solvents118 and water119 in the interlayer have been
studied at very close approach.
(B) Atomic Force Microscope: The development of the
AFM120 dramatically expanded the types of materials for which
direct intersurface force measurements121 could be determined.
The AFM has permitted the study of DLVO forces,122 such as
EDL and hydration.123 The opportunity of AFM to measure the
LD forces with vacuum or liquid interlayers is complicated,
because the complex shape of the tip affects the analysis of the
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
2133
now been done for TiO2,127 gold,128 Al2O3,129 ZrO2,130,131 ice,132
and other materials.
Fig. 16. Parametric tip model. R is the tip radius, rmax the radius of the
cylindrical part of the probe, ␥ the cone angle, ␣ ⫽ ␲/2 – ␥ the angle
included in the spherical cap, z0 the probe–sample separation, r the radius
at any point on the surface, x the distance from the cone apex, and a, b, and
c define the spherical cap section and the conical section of the probe.
force– distance relation to extract the Hamaker constant. This is
where the simple crossed cylinders geometry of the SFA is an
advantage. Initial work by Hartmann124 and others and recent
application of the intersurface stress tensor approach (discussed in
more detail in Section III(6)) now permits the extraction of
Hamaker constants from measured force– distance relations,125
assuming the LD forces dominate. The parametric tip force–
distance relation (Eq. (54)) permits the nonlinear fitting of the
measured LD force to the Hamaker constant ANR and the geometrical parameters of a conical AFM tip, with a spherical cap (shown
in Fig. 16).
F z共 z 0兲 ⫽
AR 2 共1 ⫺ sin ␥兲共R sin ␥ ⫺ z 0 sin ␥ ⫺ R ⫺ z 0 兲
6z 20 共R ⫹ z 0 ⫺ R sin ␥兲 2
(6) Adhesion and Sintering
The LD forces can be approached in different ways, e.g., at the
discrete point-to-point approach of Hamaker for atoms and molecules or at the continuum level; the continuum approach of Lifshitz
is the most useful for the formulation and analysis of “macroscopic” problems. There are two important macroscopic manifestations
of molecular interactions: interactions within a body that result in
surface tension and interactions between bodies that affect phenomena such as adhesion and the behavior of colloids. Often, only
one of the macroscopic effects of molecular interactions is considered. For example, in the cohesive zone theory of fracture
mechanics,133 forces behind the crack tip are assumed to be based
on interactions across the gap. As such, the forces are oriented
normal to the free surface. On the other hand, forces at a sintering
neck, which is essentially an exterior crack, usually have been
modeled based on surface tension and local curvature.134 The
difference between the two is illustrated in Fig. 17. Both sets of
forces are energetically consistent with the intermolecular forces
on which they are based in that both yield the same work per unit
extension of surface area. However, because they are so different
in detail, they lead to very different kinetics. In many problems,
one of the macroscopic effects of molecular interaction is sufficient to model surface-force-driven processes. However, in the
problem of viscoelastic sintering, viscous sintering of fine particles, and viscoelastic crack growth, we must consistently describe
the in-plane surface tension and the normal dispersion force or
“interaction across a gap.”135
The continuum treatment of surface tension yields the notion of
surface stress. For liquids, the associated pressure p on a surface is
proportional to curvature ␬ via Eq. (55), developed by Young and
Laplace,136
p ⫽ ␥␬ ⫽ ␥
⫺A tan ␥共 z 0 sin ␥ ⫹ R sin ␥ ⫹ R cos 2␥兲
⫹
6 cos ␥共 z 0 ⫹ R ⫺ R sin ␥兲 2
(54)
AFM tips are most commonly fabricated from Si3N4, and many
studies of intersurface forces in Si3N4 have been reported.126 By
the attachment of a colloidal particle of a material to the AFM tip,
the ability to measure intersurface forces between many colloidal
particles and an arbitrary substrate material has become one of the
most fruitful applications of the AFM in colloid science. Intersurface force measurements in aqueous or nonaqueous systems have
冉
1
1
⫹
R1 R2
冊
(55)
where R1 and R2 are the principal radii of curvature. Two different
approaches have been followed in the continuum treatment of
interactions across a gap. The more general is that of Lifshitz et
al.,19 in which interactions are calculated in terms of virtual
photons and electromagnetic fluctuations and interactions between
half spaces. This approach relates interaction forces to continuum
properties of the interacting materials, including those that fill the
gap.
The Lifshitz approach is based on the recognition that the use of
the point-to-point interaction energy is too simplistic to compute
Fig. 17. Schematic illustration of contact growth modes in elastic adhesion, which is driven by direct attraction across the gap ahead of the contact neck,
and viscous sintering that is driven by tractions that are modeled as being proportional to the surface curvature. For elastic adhesion, the contact grows in
a zipping mode, whereas, for viscous sintering, the contact grows in a stretching mode.134
2134
Journal of the American Ceramic Society—French
interactions between macroscopic solid bodies of atoms or molecules with a high density of packing. Not incorporating this fact,
Hamaker’s approach12 postulates that the total interaction between
solid bodies can be computed by integrating pairwise interactions
between infinitesimal individual regions, and these interactions are
assumed to be central, i.e., aligned along the vector connecting
them. The power of this assumption is that it is possible to
calculate interactions between various-shaped bodies. For interactions between identical materials with no intervening medium
(vacuum), the functional form of the interaction is correct, but the
Hamaker constant cannot be related directly to the intermolecular
interaction energy.137 There are several other cases where the
Hamaker assumption of pairwise additivity breaks down completely, i.e., in computing interactions between two materials with
a third intervening medium. The Lifshitz approach has been
detailed in Section II(2); in the following paragraphs we develop
the pairwise additive approach for interaction between bodies of
general shape.
Let the pairwise interaction energy of two molecules as a
function of distance d be w(d). The interaction energy e(d) between
two elemental volumes dV1 and dV2 then is
e共d兲 ⫽ ␳ 1 ␳ 2 w共d兲 dV 1 dV 2
(56)
where ␳1 and ␳2 are the number densities of molecules. These
infinitesimal volumes interact, each with every other, and it is
assumed that the total energy of the system is the sum of individual
interaction energies. The interaction energy E between two bodies
defined by the volumes V1 and V2 is then given by the double
volume integral
E⫽
冕冕
V2
␳ 1 ␳ 2 w共d兲 dV 1 dV 2
(57)
V1
Similarly, the total force of interaction between two bodies can be
computed.
F̂ ⫽
冕冕
V2
␳ 1 ␳ 2 f̂共d兲 dV 1 dV 2
(58)
V1
where f̂(d) is the interaction force given by the spatial gradient ⵜ(䡠)
of the interaction energy as
f̂共d兲 ⫽ ⫺ⵜw共d兲
(59)
Vectors are denoted by a caret (e.g., f̂) and second rank tensors
ˆ Interactions handled in this manner have
by a double caret (e.g., ĥ).
been computed for various geometries.29 If
w共d兲 ⫽ ⫺
A NR
␲ ␳ 1␳ 2d 6
2
(60)
which is the vdW interaction energy for point-to-point interactions,
then the interaction energy of two half spaces separated by a
distance d reduces to Eq. (2) where m ⫽ 2 and ANR is the Hamaker
constant.
The application of the point-to-point summation approach
remains difficult for arbitrarily shaped bodies. Derjaguin138 introduced an approximation that is accurate when surface radii of
curvature are large compared with the gap between the bodies. It
consists of treating opposing surfaces as composed of elements
that interact as would half spaces. Formally, surface traction T̂, a
force per unit area, is then computed as
T̂ ⫽ F共d兲共q̂ 䡠 n̂兲q̂
Vol. 83, No. 9
Body 1 causes distributed forces b̂ (per unit volume) in body 2,
given by,
b̂ ⫽ ␳ 1 ␳ 2
冕
ⵜ 2 共w共d兲兲 dV 2 ⫽ ␳ 1 ␳ 2
V2
冕
w共d兲n̂ 2 dS 2
where dV is a volume element and dS a surface element.
Following a special integration procedure, these can be replaced
by effective surface tractions T̂ on the surface of body 2. Then,
the influence of body 1 is represented by an intersurface stress
tensor ĥˆ such that
T̂ ⫽ hˆˆ 䡠 n̂ 1
hˆˆ ⫽ ␳ 1 ␳ 2
(63)
冕
n̂ 2 Ĝ dS 2
(64)
S2
where Ĝ is the vector field
Ĝ ⫽ 共x̂ 2 ⫺ x̂ 1 兲v共d兲
(65)
and x̂1 and x̂2 are position vectors to surfaces 1 and 2. The scalar
function of distance between the two surfaces, v(d), is derived
from the intermolecular interaction energy:
v共d兲 ⫽
1
d3
冕
⬁
w共t兲t 2 dt
(66)
d
ˆ
Therefore, the external influence of body 1 is represented by ĥ.
ˆ it produces a
If another surface is introduced into the field of ĥ,
traction on that surface in a manner analogous to the internal stress.
The total force of interaction on body 1 because of body 2 is equal
and opposite to that on body 2 because of body 1. However, unlike
in Derjaguin’s approximation, the distribution of forces on the two
bodies may be very different.
It has been pointed out140 that the stress ĥˆ is formally equivalent
to the Maxwell stress of electromagnetic theory,141 in that its
divergence yields the distributed body force (or force per unit
volume). Using this construction, an alternative method for solving
this class of problems has been developed.
As mentioned earlier in this section, sintering is usually analyzed using only the surface tension representation (Fig. 17(a)) of
surface forces.142,143 However, based on the Johnson–Kendall–
Roberts theory of adhesion between elastic spheres,144 which
neglects the surface tension aspect but includes the attraction
across a gap (Fig. 17(b)), all sintering begins by the formation of
a neck due to elastic deformations, the so-called zeroth stage of
sintering.142 Both manifestations of surface forces act simultaneously at a sintering neck. Moreover, there may be a regime of
particle size in which sintering is controlled by direct attractive
forces across the gap ahead of the neck, rather than by surface
tension.145 By combining both surface tension and direct attractive
interactions, Jagota et al.134 have studied the problem of sintering
of a Maxwell viscoelastic sphere to a rigid half space. Their
analysis retrieves the known limits: initial elastic adhesion driven
by direct attractive forces and final contact growth driven by
surface tension. For submicrometer particles, a new mode of
viscous sintering develops, which is driven by direct attractive
(61)
where F(d) is the interaction force (per unit area) between two half
spaces, d the distance between the surfaces measured along q̂, and
n̂ the surface normal.
Hamaker’s approach for the computation of interactions between bodies has been generalized recently to account for arbitrary
geometry.135,139 Consider two bodies 1 and 2 as shown in Fig. 18.
(62)
S2
Fig. 18.
Two arbitrarily shaped bodies.137
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
forces. In it, the contact “zips” shut in contrast with the conventional sintering calculations, where the contact is stretched open.
Classical Herring scaling between sintering time and particle
radius breaks down as a new length scale, the range of cohesive
forces, enters into the relationship. Scaling of contact radius with
time,146,147 instead of t1/2 given by the Frenkel model, is between
t1/5 and t1/7. These predictions remain to be verified experimentally
for viscous sintering; however, the zipping mode of contact growth
explains well the observed sintering kinetics in polymeric systems
that have more-complex rheology.148
(7) Equilibrium Intergranular and Surficial Films
The properties of liquid-phase sintered or liquid-infiltrated
materials depend on the characteristics of the intergranular phases
and associated interfacial films (as shown in Fig. 19) as well as the
morphologies of the bulk phases.149,150 Many studies have pointed
out that the amount and composition of the sintering aids influence
the microstructure evolution during sintering,151–154 thereby affecting the chemical and physical properties of the final microstructures. The actual interfacial features also strongly influence
mechanical and physical properties,155 e.g., creep resistance156,157
or fracture toughness158 –162 of Si3N4- or SiC-based materials, or
the electrical properties in thick-film resistors,163 varistors,164 or
high-Tc superconductors.165 In all these cases, the presence of
intergranular films and the specific impurities in them profoundly
alter macroscopic properties, often beneficially. Thus, it is important that the actual composition and bonding of the interfacial films
can differ from those of the associated bulk glass phase.163–171 The
factors controlling the composition and structure of these films are
a matter of current research, but the dispersion forces across them
are one of the important factors controlling their stability.
Numerous investigations have focused on features affecting the
sintering process of materials with low liquid contents. Densification is frequently expedited by the presence of a small liquid
fraction. The driving force is usually described in terms of the
effect of capillary pressure on the grain-boundary formation
process.172–174 The capillary forces contemplated result from the
menisci at liquid–vapor interfaces between contacting particles
that drive sintering by pulling particles together. An important
concept is that, despite the compressive force induced across
particle contacts, enough of the liquid-forming material must
remain adsorbed at the boundaries to provide the high boundary
diffusion required to account for enhanced rates of densification
via grain-boundary formation.171,175 In Si3N4-based materials,
boundaries with 1–2 nm thick, amorphous oxynitride films usually
form between particles during liquid-phase sintering.176 –178 Moreover, it has been shown that the boundary thickness is almost
constant in a given material although it adjusts depending on the
Fig. 19. HRTEM image of an equilibrium intergranular film in
Si3N4.155,179
2135
exact composition,155,179 –181 implying that the thickness reflects
some equilibrium condition.182,183 However, under conventional
sintering conditions in some systems, e.g., ZnO–Bi2O3, the capillary forces reversibly influence the grain-boundary composition;
hence, it is not only a result of an equilibrium among local forces
acting across a boundary or between two solid–liquid interfaces.170,184
Complementary approaches to probe the particle interactions
and the roles of interfacial composition begin with considering the
solid phase (either powder or dense polycrystal, initially) being
immersed in a massive amount of liquid at high temperature, as
demonstrated by Shaw and others.185–188 Such systems largely
evolve in the absence of the capillary forces induced by liquid–
vapor interfaces, under conditions where dispersion forces, and
also surface energies, can be dominant. Consider when a small
amount of refractory particles is immersed at high temperature in
a liquid matrix: A dilute suspension is realized. This can be treated
as a colloidal system, where local forces among particles govern
their mutual interactions. A wide range of behaviors can prevail in
which the specific particle–particle interaction influences the
degree of interconnectedness and the types of grain boundaries that
develop between particles.188,189 These features then influence the
microstructural evolution via the strength of the solid network, as
well as through the interfacial kinetic parameters. Alternatively, if
a dense polycrystalline material is immersed in excess liquid at
high temperature, liquid penetration along the triple points and
possibly through grain boundaries can eventually occur, as has
been observed for Al2O3 in silicates.185,188
If a more global understanding of the underlying interparticle
forces, interfacial reactions, and kinetics emerges, it should be
feasible to design and fabricate a richer variety of materials with
superior properties. For example, the boundary penetration process
has profound implications concerning the corrosion or oxidation
resistance of ceramics, and it can be beneficially used to form
functionally graded materials.188 Graded materials made with
Al2O3 or Si3N4 partially penetrated by silicates have been shown
to have enhanced resistance to wear and contact damage.190,191
Moreover, based on this principle, a transient liquid can be used to
promote brazing or diffusion bonding between pieces of ceramic
leading to seamless joints,192,193 where the liquid is imbibed into
the bulk ceramic pieces.
(A) Various Oxide Systems: In many ceramic systems sintered with various impurities and dopants, a liquid phase exists at
temperature and is often manifest by residual glassy phases or
grain-boundary films in the as-cooled materials. Evidence of
continuous silicate-rich or other amorphous phases and films have
been seen in a wide variety of oxide,194,195 nitride,196,197 and
carbide160 ceramics. These include materials, such as MgAl2O4, in
which the fluoride-rich sintering aid leads to a continuous boundary phase.198 Recently, it has been appreciated that the amorphous
grain-boundary films in some systems have an equilibrium thickness, which indicates that these are adsorption films that formed in
equilibrium with some second phases at temperature. More recently, it has been found that these films have composition
different from the high-temperature liquids with which they were
in equilibrium. Thermodynamically, these films can be described
in terms of Gibbsian adsorption.105 However, an atomistic description can be considered in terms of a force balance across the film,
which must involve gradients across the film as well as differences
in composition between the film and the second-phase regions.
This fundamentally includes the dispersion forces across the film.
Simpler descriptions, which have a strong parallel with lowtemperature colloidal theories, have included a variety of disjoining forces with the assumption that the primary attraction force is
the dispersion force. The neglect of other short-range attractive
forces may be an important deficiency, but, for thicker films, it is
likely that the dispersion force is the largest attractive force.
Thus, based simply on trends in dispersion forces (Table VIII),
it has been expected that multilayer boundary films should be
stable and thick for grains of low-index oxides, such as MgO or
Al2O3, together with silicate liquids.182 Indeed, for Al2O3, there
are many observations of extensive glassy pockets and films in
2136
Journal of the American Ceramic Society—French
Vol. 83, No. 9
Table VIII. Nonretarded Hamaker Constants for Various Ceramic Systems†
Solid
no
Film
MgO
Al2O3
Si3N4
TiO2
Pb2Ru2O7
ZnO
SrTiO3
1.70
1.75
2.02
2.56
3.56
1.94
2.3
SiO2
SiO2
SiO2
SiO2
SiO2
SiO2
SiO2
†
zJ is zeptojoule (10⫺21 J).
‡
Reference 23.
ANR (zJ)
8‡
19‡
33§
40‡
28‡
9††
⬃45‡‡
§
Reference 234.
ANR (zJ)
Film
3‡
10‡
21§
26‡
31‡
3††
⬃30‡‡
Pb-Al-SiOx
Pb-Al-SiOx
Pb-Al-SiOx
Pb-Al-SiOx
Pb-Al-SiOx
Pb-Al-SiOx
Pb-Al-SiOx
¶
Reference 235.
polycrystals.194,199 –203 Moreover, for Al2O3 (both pure and containing various amounts of CaO, SiO2, or other impurities)
extensive penetration of dense polycrystals by silicate liquids
occurs, leading eventually to widespread disruption, i.e., deflocculation of the grains, which may mean the dihedral angle is low
or the liquid is virtually wetting.185,204 Al2O3 powders initially
mixed into glasses give similar results, in which particles remain
widely separated, but do concentrate somewhat while settling
because of gravity.205 However, for Al2O3 and MgO, which can be
extensively penetrated by silicate liquids,206 a discernable fraction
of boundaries is not wetted or obviously penetrated. The condition
of these boundaries, i.e., whether moist, dry, or special, has not
been assessed by analytical or high-resolution TEM. However,
TEM studies of Al2O3 polycrystals with small levels of glass
indicate that the dry boundaries are typically special, often
twins.199,200,203 Indeed, there has been comparatively little characterization of the actual films in most studies of these two oxides.
Similarly, it was found that iron-doped SrTiO3 polycrystals
having excess TiO2 exhibited similar ⬃1 nm disordered grain
boundaries containing approximately a monolayer of excess titanium and half that of iron in the films.207,208 Moreover, earlier
TEM studies of quenched samples of less-pure ZnO–Bi2O3209,210
and TiO2-rich SrTiO3211 revealed that each has a wetting transition
for which ␺ 3 0 above a critical temperature, ⬃1150° and
1450°C, respectively. The transition temperatures were reduced
with other impurities present for SrTiO3.211 Rather similar results
were recently seen in Al2O3 doped with CaO–SiO2. The grain
boundaries were wide disordered layers, yet contained only about
a monolayer of adsorbed impurity, and the Ca:Si ratio was much
higher than in the liquid pockets.171 Obviously, these results imply
caution in interpreting earlier studies of oxides containing glassy
impurities.
(B) Intergranular Films in Si3N4: (a) Overview: Numerous
investigations of Si3N4, have consistently revealed that a very high
fraction of grain boundaries are composed of an amorphous
silicate film that is considered to have an equilibrium thickness,
and the film thickness depends as much or more on the composition of the system as on the crystallography of the boundary.155,180,181 Analytical TEM studies show that films often
contain more nitrogen than do the pockets and can have different
concentrations of cations.166 –168 Moreover, for Si3N4 grains equilibrated with a RE-aluminum silicate (where RE is one of the
yttrium series or various rare earths), the mean film thickness
depends on the specific rare earth used, but it is virtually the same
for polycrystals with 5%–12% liquid by volume and for small
clusters of grains found in dilute suspensions of powder in
glass.179,181 Apparently, a particle–particle attraction exists in this
system that is roughly comparable to the capillary forces often
considered to drive sintering. However, in parallel, the ␣ to ␤
transformation and subsequent particle growth (Ostwald ripening)
occur via diffusion through the liquid and films. The rates are
highly anisotropic, with preferred growth parallel to the c-axis,
yielding long grains with strongly faceted prism faces.212
(b) In Situ Dispersion Force Determination by SR-VEELS:
We have shown how to calculate full spectral Hamaker constants
from experimental interband transition strengths. Quantitative
analysis of SR-VEEL spectra and SR-VEEL spectrum images can
now supply the required interband transition strength results for
††
Reference 170.
‡‡
Film
Y-Al-Si-O-N
10TiO2䡠90SiO2
Bi2O3–ZnO eut.
ANR (zJ)
8§
25¶
4‡‡
Estimated.
individual intergranular films (IGFs) in Si3N4. The combination of
SR-VEEL spectrum imaging and full spectral Hamaker constants
permits the in situ determination of vdW forces on the IGFs in
Si3N4-based ceramics and on the local variations in dispersion
forces throughout the microstructure of individual Si3N4 samples.24
The Hamaker constants representing the LD forces were determined from in situ SR-VEEL spectroscopy for three different IGF
chemistries in Si3N4 material. Full profiles from one Si3N4 grain
across the amorphous IGF into the grain on the other side were
acquired with a new spectrum image acquisition system. The full
data set was analyzed, and the interband transition strength
determined as a function of beam position. From those data, the
Hamaker constant was calculated for the actual interface investigated, based on spectra from the center of the IGF.
The samples studied fall into two classes:24 those based on
lanthanide (R) glasses of the type R-Si-Al-O-N and on simple
silicate glass compositions containing calcium of the type Ca-SiO-N. The lanthanide glass samples used either yttrium-aluminum(YAl-) or lanthanum-aluminum- (LaAl-) doped silicon oxynitride
(SiON) glass.212 The calcium-doped SiON glass–Si3N4 samples
were made by Tanaka.180,213 The oxygen content was 1.3 wt% and
the calcium concentration was 450 ppm, with no other cation
impurities detected.
SR-VEEL spectra were acquired on the Si3N4 samples with a
parallel EEL spectroscopy system fitted to a dedicated scanning
transmission electron microscope (STEM) operated at 100 keV.
The incident electron beam spot size was 0.6 nm diameter, and the
energy resolution of the whole system was better than 0.7 eV. Each
spectrum covered the energy-loss range from –20 to 80 eV. Figure
20 shows the microstructure of one of the interfaces analyzed in
the LaAl–Si3N4 material.
The LD spectra ε2(␰) were calculated and then accumulated in
a spectral database from which any combinations of them could be
used to calculate the Hamaker constants of interest. For each
VEEL spectrum image, three interband transition strengths were
selected, one from each Si3N4 grain and one from the middle of the
IGF (Fig. 21). Full spectral Hamaker constants for ANR
[Si3N4兩vacuum兩Si3N4] and ANR [Si3N4兩IGF兩Si3N4] were then determined from each SR-VEEL spectrum image in each type of
Si3N4 material.
At the end of the analysis of the SR-VEELS data, the index of
refraction can be calculated from the dielectric function across the
IGF (Fig. 22). The index for bulk Si3N4 is recovered, but the index
of refraction for the IGF also is obtained by this method.
The indexes of refraction for the IGFs and the Hamaker
constants show a dependence on the composition of the IGF. When
the Hamaker constant is plotted against the IGF index of refraction
(Fig. 23), it is observed that the Hamaker constant decreases
almost linearly with increasing index of refraction of the IGF.
From the profiles of the interband transition strength across the
IGF, it is possible to measure an IGF thickness. This is not
identical to the IGF thickness as determined by HREM but is
related to it. Here it serves as an indication of the variation in IGF
thickness between different interfaces in a particular material. A
definite correlation between this IGF thickness and the Hamaker
constant exists for the lanthanum–aluminum material (Fig. 24).
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
2137
Fig. 22. Index of refraction determined from the index sum rule applied
to the interband transition strength spectrum image from LaAl–Si3N4.
(Reprinted with permission from Elsevier Science.24)
The variation of the dispersion forces on the IGFs in these three
types of Si3N4 are shown in Fig. 25 and vary from 2 to 23 zJ, a
factor of ⬎10 in the Hamaker constant. Previously, there may have
been speculation that variations in the dispersion force were not
important in dictating the force balance. However, a variation by a
factor of 10 in the dispersion force might well be an important
component in the force balance in optimized Si3N4 materials
exhibiting equilibrium IGFs. If the dispersion force is the major
attractive force counterbalancing the other repulsive terms, then
changes of this magnitude in the attractive term are critical.
Alternatively, the dispersion force variation might become negligible in comparison to other terms that are of larger absolute
magnitude, with the effect that the dispersion force would not be
important. However, the present results discount this idea. From
the study of individual IGFs in LaAl–Si3N4, a direct correlation
between variations in the dispersion force on the IGF and the
measured IGF thickness is found. This demonstrates that for the
optimized Si3N4 materials studied here, which have been sintered
to a final state microstructure and exhibit equilibrium IGFs, a
major contributing term in the IGF thickness is the LD force on the
IGFs. The IGF thicknesses in various optimized Si3N4 samples
with common glass systems correlate very closely to the magnitudes of the Hamaker constants of, and dispersion forces at, the
IGFs. From HR-TEM measurements, the average IGF thicknesses
for these materials are 1.2, 1.45, and 1.6 nm for YAl-, Ca-, and
LaAl-doped SiON glass–Si3N4 materials. Considering the
lanthanide-doped glass system, the LaAl-doped system has lower
values of the dispersion force for three of the four interfaces, and
this material exhibits thicker intergranular films. At the same time,
variations of the dispersion force within a single material are also
apparent and can be correlated to the variation of the equilibrium
film thicknesses from their average values for the material.
The SR-VEELS method of in situ Hamaker constant determination is accurate and reproducible with comparison to VUV
measurements for the bulk materials and repeated measurements
on numerous individual IGFs. For various glass additives having
various film thickness, the accurate Hamaker constants provide a
basis to calculate the first term in the controlling force balance (Eq.
(53)). Local variations in Hamaker constants within the microstructure of a single sample correlate inversely to the distribution
of IGF thickness observed. For these optimized Si3N4 materials,
Fig. 21. Interband transition strength spectra for the IGF and each Si3N4
grain from LaAl–Si3N4. (Reprinted with permission from Elsevier Science.24)
Fig. 23. Hamaker constant ANR
121 as a function of the index of refraction of
the IGF for all the boundaries studied. (Reprinted with permission from
Elsevier Science.24)
Fig. 20. STEM micrographs of a LaAl–Si3N4 sample showing where
SR-VEEL spectrum images were acquired. The 10 nm spectrum image (a
line along which the multiplexed SR-VEEL spectrum image was acquired)
is shown. (Reprinted with permission from Elsevier Science.24)
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Journal of the American Ceramic Society—French
Vol. 83, No. 9
ANR
12321). As the interlayer becomes more complex, the interaction
energy becomes dependent on the particulars of the structural
variation of the interlayer. Systems more complex than ANR
12321 are
often seen in biology, where the behavior of complex many-layer
films result in cell walls and cell– cell interactions. Parsegian43,221,222 has developed a solution for the LD interaction for
multilayered interlayers, but only in the approximation that the
center-most single layer has variable thickness. This is the same
implicit approximation that was used for analysis of the soap
film42 and may not have applicability in many ceramic systems.
Additional aspects of the interfacial properties of graded interfaces, beyond dispersion, have been discussed recently in the
literature. For example Ackler et al.223 have investigated the
effects of amorphous, thin intergranular film that abuts crystalline
surfaces with differing orientation. They use a diffuse interface
method to calculate the variation of order that is induced by the
Fig. 24. Hamaker constant ANR
121 for LaAl–Si3N4 material as a function of
the IGF thickness as measured from the SR-VEEL index of refraction
profiles across the IGF.
Fig. 25. Hamaker constant ANR
121 for [Si3N4兩IGF兩Si3N4] for each interface
studied in the LaAl-, YAl-, and Ca-doped SiON glass–Si3N4 materials.
the dispersion forces vary over a range from 2 to 12 zJ, corresponding to a 50% increase of the IGF thickness. This ability to
experimentally determine Hamaker constants in situ represents an
important new tool for dispersion force and wetting studies.
(C) Intergranular and Surficial Films in ZnO: The ability to
rigorously calculate Hamaker constants for inorganic systems has
facilitated the study of interfacial phenomena in many other
ceramics. Chiang and co-workers170,214 –216 have systematically
studied the model varistor system Bi2O3-doped ZnO, and they
have determined that the equilibrium state of nonspecial grain
boundaries is a disordered, bismuth-enriched film of 1–1.5 nm
thickness (Fig. 26), not unlike the siliceous films found in other
oxides and Si3N4. Moreover, the bismuthate film coexists with
bulk solid or liquid phases, and it has been argued to be a true
surface phase whose thickness and composition are determined by
dispersion forces among other interactions.215 These intergranular
films have been implicated in subsolidus-activated sintering217 as
well as varistor behavior.218
Recently, surface amorphous films of stable, nanometer thickness also have been identified in Bi2O3–ZnO (Fig. 27) and other
oxides,219,220 the quantitative interpretation of which requires
evaluation of the ANR Hamaker constant for surface layers, a
quantity that can be positive or negative, acting to thin or thicken
a surface film.
(D) Diffuse Interface Theory: The properties of diffuse or
graded interfaces are of general interest in ceramic systems. For
the LD interaction, we have discussed the solutions for (ANR
123 and
Fig. 26. Intergranular film, shown at a nonspecial grain boundary, in a
model varistor system of Bi2O3-doped ZnO. Interlayer film is a disordered,
bismuth-enriched film of 1–1.5 nm thickness.
Fig. 27. Amorphous surficial film of stable nanometer scale thickness
seen in Bi2O3-doped ZnO.
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
2139
presence of each crystalline surface. Their results indicate that,
depending on misorientation of the crystals, the induced order can
produce positive or negative steric forces on the thin amorphous
film. The theory depends on measurable bulk thermodynamic
properties. Bobeth, Clarke, and Pompe224 have published a diffuse
interface theory that treats composition gradients between triple
junctions and adjoining thin-film regions. They calculate an
effective osmotic force that accounts for observed variations in
film thickness with solute addition. Warren, Carter, and Kobayashi225 have formulated a diffuse interface theory that includes
crystalline orientation and induced interfacial order as well as
composition as order parameters. This method has been used to
calculate microstructural evolution for systems of many crystallites growing in an amorphous phase. Thin intergranular films have
been observed in these simulations but have not been characterized.
(E) When Is an IGF Too Thin—Continuum Approximation:
An interesting question arises in the case of very thin intergranular
films: When is the film thickness so thin that we are violating the
continuum approximation inherent in the Lifshitz approach? Or,
what is the smallest quantity of the material that has the properties
that are essentially bulklike and that have comparable atomic
structure, oxidation states, and electronic structure and bonding
and, therefore, optical properties? Because the LD interaction is
essentially electronic in nature, resulting from the interband
transitions of the electronic structure, it is the nature of the bonding
and its extent in real space that determines the intrinsic scaling
that, in turn, determines the minimum size for the continuum
approximation. Therefore, in vdW solids, such as polymeric
materials, the interatomic bonding is fully satisfied in the polymer
chain, and it is the size of the molecular moiety, such as the chain
diameter or chain length, or a few times the chain diameter, that
can be considered as the smallest length for which bulk,
continuum-like, properties would be expected. For silicates and
glass formers, this intrinsic length scale is given by the tetrahedra
size of 0.23 or 0.15 nm; therefore, the bulklike electronic structure
for silicates should be observed for silicates in the range of a few
SiO2 tetrahedra or ⬃0.6 nm in size.226 For materials with extensive three-dimensional bonding, the intrinsic length necessary to
establish the bulklike electronic structure may be larger. For
particular materials of interest, this question is answered by
comparing bulk band structure calculations with electronic structure calculations for clusters of increasing size. The cluster size
necessary to realize a bulklike electronic structure represents the
size necessary to satisfy the continuum approximation of Lifshitz
theory. It also can be determined by considering the real space
extent of the individual electronic wave functions in the material,
as can be determined from electronic structure calculations.227 The
intrinsic length scale for the transition to continuum behavior for
various materials is a topic that warrants additional study.
Fig. 28. YAl–SiON intergranular films in Si3N4, showing normal retardation effects corresponding to a nonwetting condition, as shown in (a) the
interaction energy (arbitrary units) and (b) the retarded Hamaker constant.
Fig. 29. YAl–SiON surficial films in Si3N4, showing normal retardation
effects corresponding to a nonwetting condition, as shown in (a) the
interaction energy (arbitrary units) and (b) the retarded Hamaker constant.
(8) Retardation of Dispersion Forces
(A) Normal Retardation Effects—Reduction in the Power-Law
Dependence: If we consider the case of the intergranular films in
Si3N4 first discussed in Section III(7)(B)(b), we can calculate the
LD or vdW interaction energy and the retarded Hamaker spectrum
for AR
121 for material 1 (Si3N4) and material 2 (the YAl–SiON
glass) shown in Fig. 28. The energy minimum at zero interlayer
thickness shows that this is an attractive dispersion force. The
nonretarded Hamaker constant is 8, but, with increasing interlayer
thickness, the Hamaker constant monotonically decreases. At a
film thickness of 10 nm, the Hamaker constant has been decreased
by a factor of 2. This case is an example of “normal retardation”;
the outcome of the retardation is a decrease in the power-law
dependence.
Let us consider an example of normal retardation in a surficial
film. A surficial film of the same YAl–SiON glass on a Si3N4
substrate gives an interaction energy, shown in Fig. 29(a), that
reaches its minimum at infinite film thickness, representing a
wetting film condition in which the film would, if able to approach
equilibrium, approach infinite thickness. The Hamaker constant,
shown in Fig. 29(b), is negative, demonstrating the repulsive force
between the air– glass interface and the glass–Si3N4 interface. As
the surficial film thickens, the Hamaker constant decreases monotonically from a nonretarded value of 17 zJ to 8 zJ at 15 nm.
(B) Equilibrium Surficial Films: The LD interaction can
result in equilibrium surficial films in systems, such as water on
ice, that exhibit an equilibrium film thickness, typically on the
order of 4 nm. The appearance of equilibrium surficial films is
very different from the normal effects of LD forces on surficial
films, where either a repulsive interaction produces wetting films
or an attractive interaction produces dewetting films. To understand the occurrence of equilibrium thickness surficial films of, for
example, water on ice, we need to understand the effect of
retardation on the magnitude of the vdW dispersion forces and the
direct effect of film thickness in modifying these forces because of
the effects of retardation.
(a) Water on Ice: The direct cause of the equilibrium surficial films of water on ice seen at temperatures to 40°C below the
2140
Journal of the American Ceramic Society—French
melting point is the change in the electronic structure of H2O
between the liquid and solid states. Consider the interband transitions of water and ice shown in Fig. 30.87,88 Although the
bandgaps of water (n ⫽ 1.41) and ice (n ⫽ 1.31) are comparable,
there are some differences in the interatomic bonding in the region
of 10 –16 eV, and a dramatic difference in the region of 24 –50 eV.
The large set of transitions in ice results from bonding between the
H2O molecules in the solid, crystalline state. The introduction of
these interatomic bonds at high energy leads to a strong attractive
dispersion force at close approach, but, as the surficial film
thickness increases, these high-energy bonds are the first to
dephase. Another interesting spectral behavior is shown in Fig. 31,
where the LD spectra of water and ice cross in the vicinity of 10.5
eV. This spectral crossing in the dispersion spectra is the signature
for equilibrium surficial film formation. Calculating the retarded
dispersion interaction (Fig. 32) shows that the interaction energy
does not exhibit the monotonic variation of the examples of normal
retardation, but, rather, there is an energy minimum for the system
at 3 nm. The retarded Hamaker spectrum is also not monotonic,
but, instead, has a zero crossing from negative, repulsive forces at
close approach to positive, attractive forces at larger film thicknesses. Therefore, for film thicknesses ⬍3 nm, the dispersion force
tends to thicken the film, whereas, for film thicknesses ⬎3 nm, the
film tends to thin under the attractive dispersion forces.
(b) Lorentz Oscillator Doping for Equilibrium Surficial Films:
The high-energy transitions of the substrate produce the LD
spectrum crossing, thereby causing the formation of equilibrium
surficial film formation for water on ice. A more practical method
of designing materials with surficial films is to dope the wetting
film material, leaving the substrate unchanged.228 Figure 33 shows
the interband transitions for an Al2O3 substrate and a wetting film
of SiO2. From the perspective of the LD interaction and considering the index of refraction, SiO2 (n ⫽ 1.51) wets Al2O3 (n ⫽
1.75). Using a Lorentz oscillator (LO) centered at 12 eV, we can
add a set of interatomic bonds to the interband transition strength
of the SiO2, synthesizing a new interband transition strength
spectra comparable to a doped silicate glass with an increased
index of refraction of 1.78. The LD spectra (shown in Fig. 34) for
the LO-doped SiO2 now crosses the spectrum of Al2O3. From the
index of refraction approximations, the LO-doped SiO2, whose
index is higher than that of Al2O3, does not wet the Al2O3
substrate. However, from the calculation of the retarded Hamaker
interaction, Figure 35 shows that there again is an energy minimum observed at 10 nm and a zero crossing of the Hamaker
spectrum. This approach of synthesizing new LO-doped interband
transition strengths permits the systematic study of the dispersion
interactions and their use in materials design.
Fig. 30. Interband transition strength of (blue) ice and (red) water,
showing large changes in the interatomic bonding in the frozen state.
Vol. 83, No. 9
Fig. 31. LD spectra for (blue) ice and (red) water, showing a crossing of
the LD spectra at 12 eV.
We have considered only the effects of dispersion; the behavior
of any particular material may be dominated by other interactions.
For other systems, such as hydrocarbon solvents wetting vdW
solids, the dispersion interaction would be expected to dominate.229
(C) Bimodal Wetting/Dewetting: The retardation of the LD
interaction can result in other novel wetting conditions in addition
to the equilibrium surficial films just discussed. Calculating the
retarded Hamaker interaction of the LaAl –SiON intergranular
glass (discussed in Section III(7)(B)(b)) as a surficial film on a
Si3N4 substrate produces a global energy maximum in the interaction energy shown in Fig. 36(a). This system either wets or
dewets, depending on its initial conditions. If the initial film
thickness is ⬍8 nm, then the dispersion interaction is attractive,
and the film thins. However, if the film thickness is ⬎8 nm, then
the interaction is repulsive and the film thickens. The retarded
Hamaker spectrum shows the opposite curvature, compared with
the equilibrium surficial film case, and also exhibits a zero
crossing, with the transition changing from an attractive close
approach to repulsive for thicker films. This condition corresponds
to a bimodal wetting/dewetting condition, an interesting film
instability with potentially useful applications.
Fig. 32. Water on ice, showing the appearance of an equilibrium surficial
film, as shown in (a) the interaction energy (arbitrary units) and (b) the
retarded Hamaker constant.
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
2141
Fig. 33. Interband transition strength of (red) Al2O3 and (green) SiO2
along with a (blue) synthetic spectra of SiO2 doped with the addition of a
LO to represent interatomic bonds at 12 eV.
IV.
Conclusions
The London dispersion (LD) forces, along with the Debye and
Keesom forces, constitute the long-range van der Waals forces.
London and Hamaker’s work on the point-to-point dispersion
interaction and Lifshitz’s development of the continuum theory of
dispersion are the foundations of our understanding of dispersion
forces. Dispersion forces are present for all materials and are
intrinsically related to the optical properties and the underlying
interband electronic structure of materials. The force law scaling
constant of the dispersion force, known as the Hamaker constant,
can be determined from spectral or parametric optical properties of
materials combined with knowledge of the configuration of the
materials. With recent access to new experimental and ab initio
tools for determination of optical properties of materials, dispersion force research has new opportunities for detailed studies.
There are three opportunities in dispersion force research. The first
is the development of improved index approximations that more
accurately represent nonvacuum interlayers. The second is the
development of Kramers–Kronig consistent parametric representations of the optical properties that enables improved estimation
of Hamaker constants. Expanded databases of LD spectra of
materials permit accurate estimation of nonretarded and retarded
dispersion forces in complex configurations. The third is the
development of solutions for generalized multilayer configurations
of materials that are needed for the treatment of more-complex
problems, such as graded interfaces.
Dispersion forces are critical in materials applications. Typically, they are a component with other forces in a force balance,
and it is this balance that dictates the resulting behavior. The
ubiquitous nature of the LD forces makes them a factor in a wide
Fig. 34. LD spectra for the (red) Al2O3, (green) SiO2, and (blue) SiO2
with the 12 eV LO dopant. LD spectra crosses the doped and undoped SiO2
near 10 eV.
Fig. 35. LO doping of a SiO2 interlayer material showing the appearance
of an equilibrium surficial film, as shown in (a) the interaction energy
(arbitrary units) and (b) the retarded Hamaker constant.
spectrum of problems; they have been in evidence since the
pioneering work of Young and Laplace on wetting, contact angles,
and surface energies. Additional applications include the interparticle forces that can be measured by direct techniques, such as
atomic force microscopy. LD forces are important in adhesion and
in sintering, where the detailed shape at the crack tip and at the
sintering neck can be controlled by the dispersion forces. Dispersion forces have an important role in the properties of many
ceramics that contain intergranular films, and here the opportunity
exists for the development of an integrated understanding of
Fig. 36. Surficial film of LaAl–SiON glass on Si3N4 substrate showing
bimodal wetting/dewetting, as shown in (a) the interaction energy (arbitrary units) and (b) the retarded Hamaker constant.
2142
Journal of the American Ceramic Society—French
intergranular films that encompasses dispersion forces, segregation, multilayer adsorption, and structure. The intrinsic length scale
at which there is a transition from the continuum perspective
(dispersion forces) to the atomistic perspective (encompassing
interatomic bonds) is critical in many materials problems, and the
relationship of dispersion forces and intergranular films may
represent an important opportunity to probe this topic.
The LD force is retarded at large separations, where the transit
time of the electromagnetic interaction must be considered explicitly. Novel phenomena, such as equilibrium surficial films and
bimodal wetting/dewetting, can occur in materials systems when
the characteristic wavelengths of the interatomic bonds and the
physical interlayer thicknesses lead to a change in the sign of the
dispersion force. Use of these novel phenomena in future materials
applications raises interesting opportunities in materials design.
Acknowledgments
The author would like to acknowledge the contributions of Anand Jagota to
Section III(6), Rowland Cannon to Section III(7)(A), Yet-Ming Chiang to Section
II(7)(C), W. Craig Carter to Section III(7)(D), and Barbara S. Brown for help
preparing the manuscript. The author appreciates the long-standing support of the
DuPont Co. The support and collaboration of Manfred Rühle and my Max-PlanckInstitut für Metallforschung collaborators, Harald Muellejans, Albert D. Dorneich,
Gerd Duscher, Wilfried Sigle, Christina Scheu, Michael J. Hoffman, and Klaus van
Benthem, are gratefully acknowledged. In addition, the collaborations with David J.
Jones, Michael F. Lemon, Stephen Loughin, Lin K. DeNoyer, Rowland Cannon,
Wai-Yim Ching, and Harold Ackler are also gratefully acknowledged. I acknowledge
David R. Clarke who first encouraged my work in this field and helpful discussions
with John Wettlaufer. The comments of the reviewers are also appreciated.
References
1
(a)B. Taylor, “Concerning the Ascent of Water between Two Glass Plates,”
Philos. Trans. R. Soc. London, 27 [538.c.xx] 131 (1712). (b)F. Hauksbee, “Account
of the Experiment on the Ascent of Water between Two Glass Plates, in a Hyperbolic
Figure,” Philos. Trans. R. Soc. London, 27 [539.c.xiv] 131 (1712). (Both of these
papers are reproduced in the preface of R. Finn, Equilibrium Capillary Surfaces.
Springer-Verlag, New York, 1986.)
2
T. Young, “An Essay on the Cohesion of Fluids,” Philos. Trans. R. Soc. London,
95, 65– 87 (1805).
3
P. R. Pujado, H. Chun, and L. E. Scriven, “On the Attribution of an Equation of
Capillarity to Young and Laplace,” J. Colloid Interface Sci., 38, 662– 63 (1972).
4
J. Thomson, “On Certain Curious Motions Observable at the Surfaces of Wine and
Other Alcoholic Liquors,” Philos. Mag., 10, 330 –33 (1855).
5
J. D. van der Waals, “The Thermodynamic Theory of Capillarity Under the
Hypothesis of a Continuous Variation of Density” (in Dutch), Verhandel. Konink.
Akad. Weten. Amsterdam, 1, 8 (1893); translation published by J. S. Rowlinson, J.
Statistical Phys., 20, 200 – 44 (1979).
6
J. W. Gibbs, Trans., “On the Equilibrium of Heterogeneous Substances,” Connecticut Academy Transactions, 3, 108 –248, 343–524 (1875–1877).
7
J. W. Gibbs, The Scientific Papers of J. Willard Gibbs: Thermodynamics. Ox Bow
Press, 1993.
8
F. London, “The General Theory of Molecular Forces,” Trans. Faraday Soc., 33,
8 –26 (1937).
9
F. London, “Zur Theorie und Systematik der Molekularkräfte,” Z. Phys., 63,
245–79 (1930).
10
Von R. Eisenschitz and F. London, “Über das Verhältnis der van der Waalsschen
Kräfte zu den Homöopolaren Bindungskräften,” Z. Phys., 60, 491–527 (1930).
11
F. London, “Über einige Eigenschaften und Anwendungen der Molekularkräfte,”
Z. Phys. Chem., B11, 222–51 (1930).
12
H. C. Hamaker, “The London–van der Waals Attraction between Spherical
Particles,” Physica, 4 [10] 1058 –72 (1937).
13
W. Sellmeier, “II. Regarding the Sympathetic Oscillations Excited in Particles by
Oscillations of the Ether and Their Feedback to the Latter, Particularly as a Means of
Explaining Dispersion and its Anomalies” (in Ger.), Ann. Phys. Chem., 147, 525–54
(1872).
14
See, for example, B. J. Hunt, The Maxwellians, Cornell History of Science.
Cornell University Press, Ithaca, NY, 1994.
15
P. J. Hahn, Oliver Heaviside: Sage in Solitude: The Life, Work, and Times of an
Electrical Genius of the Victorian Age. IEEE, New York, 1988.
16
H. B. G. Casimir, “On The Attraction of Two Perfectly Conducting Plates,” Proc.
Koninklijke Nederlandse Akademie Van Wetenschappen, 51, 793–95 (1948).
17
S. S. Schweber, QED and the Men Who Made It: Dyson, Freeman, Schwinger,
and Tomonaga, Princeton Series in Physics. Princeton University Press, Princeton,
NJ, 1994.
18
(a)S. K. Lamoreaux, “Demonstration of the Casimir Force in the 0.6 to 6 Micron
Range,” Phys. Rev. Lett., 78, 5– 8 (1997). (b)S. K. Lamoreaux, “Erratum: Demonstration of the Casimir Force in the 0.6 to 6 Micron Range,” Phys. Rev. Lett., 81,
5475–76 (1998).
19
E. M. Lifshitz, “The Theory of Molecular Attractive Forces between Solids,” Sov.
Phys. JETP, 2, 73– 83 (1956).
20
I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevskii, “van der Waals Forces
in Liquid Films,” Sov. Phys. JETP, 37, [10] 161–70 (1960).
Vol. 83, No. 9
21
I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevskii, “The General Theory of
van der Waals Forces,” Adv. Phys., 10 [38] 165–209 (1961).
22
B. V. Derjaguin, A. S. Titijevskaia, I. I. Abricossova, and A. D. Malinka,
“Investigations of the Forces of Interaction of Surfaces in Different Media and Their
Application to the Problem of Colloid Stability,” Discuss. Faraday Soc., 18, 25– 41
(1954).
23
R. H. French, R. M. Cannon, L. K. DeNoyer, and Y.-M. Chiang, “Full Spectral
Calculation of Non-Retarded Hamaker Constants for Ceramic Systems from Interband Transition Strengths,” Solid State Ionics, 75, 13–33 (1995).
24
R. H. French, H. Müllejans, D. J. Jones, G. Duscher, R. M. Cannon, and M.
Rühle, “Dispersion Forces and Hamaker Constants for Intergranular Film in Silicon
Nitride from Spatially Resolved Valence Electron Energy Loss Spectrum Imaging,”
Acta Mater., 46 [7] 2271– 87 (1998).
25
V. A. Parsegian, “Long Range van der Waals Forces”; pp. 27–72 in Physical
Chemistry: Enriching Topics from Colloid and Surface Science. Edited by H. van
Olphen and K. J. Mysels. Theorex, La Jolla, CA, 1975.
26
H. Margenau and J. Stamper, “Nonadditivity of Intermolecular Forces,” Adv.
Quantum Chem., 3, 129 – 60 (1967).
27
B. Chu, Molecular Forces. Interscience, New York, 1967.
28
H. Margenau and N. R. Kestner, Theory of Intermolecular Forces. Pergamon,
New York, 1969.
29
J. N Israelachvili, Intermolecular and Surface Forces, 2nd ed. Academic Press,
London, U.K., 1992.
30
L. Bergström, “Hamaker Constants of Inorganic Materials,” Adv. Colloid
Interface Sci., 70, 125– 69 (1997).
31
D. Langbein, Theory of van der Waals Attraction, Springer Tracts in Modern
Physics, Vol. 72; p. 139. Springer-Verlag, Berlin, Germany, 1974.
32
J. W. Mahanty and B. W. Ninham, Dispersion Forces. Academic Press, New
York, 1975.
33
E. J. W. Verwey and J. Th. G. Overbeek, Theory of the Stability of Lyophobic
Colloids. Elsevier, Amsterdam, The Netherlands, 1948.
34
D. J. Shaw, Introduction to Colloid and Surface Chemistry. Butterworth–
Heinemann, Oxford, U.K., 1993.
35
W. H. Keesom, “On the Deduction of the Equation of State from Boltzmann’s
Entropy Principle”; pp. 3–20 in Communications Physical Laboratory, University of
Leiden Supplement 24a to No. 121–132. Edited by H. Kamerlingh Onnes. Eduard
Ijdo, Leyden, The Netherlands, 1912.
36
W. H. Keesom, “On the Deduction from Boltzmann’s Entropy Principle of the
Second Virial Coefficient for Material Particles (in the Limit Rigid Spheres of Central
Symmetry) which Exert Central Forces Upon Each Other and for Rigid Spheres of
Central Symmetry Containing an Electric Doublet at Their Centers”; pp. 23– 41 in
Communications Physical Laboratory, University of Leiden Supplement 24b to No.
121–132. Edited by H. Kamerlingh Onnes. Eduard Ijdo, Leyden, The Netherlands,
1912.
37
W. H. Keesom, “On the Second Virial Coefficient for Di-atomic Gases”; pp.
3–19 in Communications Physical Laboratory, University of Leiden Supplement 25
to No. 121–132. Edited by H. Kamerlingh Onnes. Eduard Ijdo, Leyden, The
Netherlands, 1912.
38
W. H. Keesom, “On the Second Virial Coefficient for Monatomic Gases and for
Hydrogen Below the Boyle-Point”; see Ref. 36, pp. 3–9.
39
P. J. W. Debye, “Die van der Waalsschen Kohäsionskräfte,” Phys. Z., 21,
178 – 87 (1920).
40
P. J. W. Debye, “Molekularkräfte und ihre Elektrische Deutung,” Phys. Z., 22,
302–308 (1921).
41
J. H. De Boer, “The Influence of van der Waals’ Forces on Primary Bonds on
Binding Energy, Strength, and Orientation, with Special Reference to Some Artificial
Resins,” Trans. Faraday Soc., 32, 10 –38 (1936).
42
B. W. Ninham and V. A. Parsegian, “van der Waals Forces Across Triple-Layer
Films,” J. Chem. Phys., 52, 4578 – 87 (1970).
43
V. A. Parsegian and B. W. Ninham, “van der Waals Forces in Many-Layered
Structures: Generalizations of the Lifshitz Result for Two Semi-infinite Media,” J.
Theor. Biol., 38, 101–109 (1973).
44
V. A. Parsegian and B. W. Ninham, “Temperature-Dependent van der Waals
Forces,” Biophys. J., 10, 664 –74 (1970).
45
L. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media. Addison
Wesley, Reading, MA, 1960.
46
A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Methods of Quantum
Field Theory in Statistical Physics. Prentice-Hall, Englewood Cliffs, NJ, 1963.
47
H. B. G. Casimir and D. Polder, “The Influence of Retardation on the
London–van der Waals Forces,” Phys. Rev. B, 73, 360 –72 (1948).
48
H. Wennerström, J. Daicic, and B. W. Ninham, “Temperature Dependence of
Atom–Atom Interactions,” Phys. Rev. A: Gen. Phys., 60, 2581– 84 (1999).
49
W. Black, J. G. V. De Jongh, J. Th. G. Overbeek, and M. J. Sparnay,
“Measurements of Retarded van der Waals Forces,” Trans. Faraday Soc., 56,
1597– 608 (1960).
50
M. Elbaum and M. Schick, “Application of the Theory of Dispersion Forces to
the Surface Melting of Ice,” Phys. Rev. Lett., 66, 1713–16 (1991).
51
L. A. Wilen, J. S. Wettlaufer, M. Elbaum, and M. Schick, “Dispersion-Force
Effects in Interfacial Premelting of Ice,” Phys. Rev. B: Conds, Matter, 52, 12426 –33
(1995).
52
J. S. Wettlaufer and J. G. Dash, “Melting Below Zero,” Sci. Am., 282 [2] 50 –53
(2000).
53
R. H. French, “Electronic Structure of ␣-Al2O3, with Comparison to AlON and
AlN,” J. Am. Ceram. Soc., 73 [3] 477– 89 (1990).
54
R. H. French, D. J. Jones, and S. Loughin, “Interband Electronic Structure of
␣-Al2O3 up to 2167 K,” J. Am. Ceram. Soc., 77, 412–22 (1994).
55
R. H. French, S. J. Glass, F. S. Ohuchi, Y.-N Xu, F. Zandiehnadem, and W. Y.
Ching, “Experimental and Theoretical Studies on the Electronic Structure and Optical
Properties of Three Phases of ZrO2,” Phys. Rev. B: Conds. Matter, 49, 5133– 42
(1994).
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
56
W. Y. Ching, “Theoretical Studies of the Electronic Properties of Ceramic
Materials,” J. Am. Ceram. Soc., 73, 3135– 60 (1990).
57
S. Loughin, R. H. French, W. Y. Ching, Y. N. Xu, and G. A. Slack, “Electronic
Structure of Aluminum Nitride: Theory and Experiment,” Appl. Phys. Lett., 63,
1182– 84 (1993).
58
F. Wooten, Optical Properties of Solids; p. 49. Academic Press, New York, 1972.
59
P. J. Mohr and B. N. Taylor, “CODATA Recommended Values of the
Fundamental Physical Constants: 1998,” National Institute of Standards and Technology, Gaithersburg, MD (http://physics.nist.gov/constants).
60
R. H. French, “Laser-Plasma-Sourced, Temperature-Dependent VUV Spectrophotometer Using Dispersive Analysis,” Phys. Scr., 41 [4] 404 – 408 (1990).
61
M. L. Bortz and R. H. French, “Optical Reflectivity Measurements Using a Laser
Plasma Light Source,” Appl. Phys. Lett., 55 [19] 1955–57 (1989).
62
R. H. French, H. Müllejans, and D. J. Jones, “Optical Properties of Aluminum
Oxide: Determined from Vacuum Ultraviolet and Electron Energy Loss Spectroscopies,” J. Am. Ceram. Soc., 81 [10] 2549 –57 (1998).
63
S. Loughin and R. H. French, “Aluminum Nitride (AlN)”; pp. 373– 401 in
Handbook of Optical Constants of Solids, Vol. III. Edited by E. Palik. Academic
Press, New York, 1998.
64
R. H. French, D. J. Jones, H. Müllejans, S. Loughin, A. D. Dorneich, and P. F.
Carcia, “Optical Properties of Aluminum Nitride: Determined from Vacuum Ultraviolet Spectroscopy and Spectroscopic Ellipsometry,” J. Mater. Res., 14, 4337– 44
(1999).
65
B. Johs, R. H. French, F. D. Kalk, W. A. McGahan, and J. A. Woollam, “Optical
Analysis of Complex Multilayer Structures Using Multiple Data Types;” pp.
1098 –106 in Optical Interference Coatings, Vol. 2253. SPIE, Bellingham, WA, 1994.
66
R. de L. Kronig, “On the Theory of Dispersion of X-rays,” J. Opt. Soc. Am., 12,
547–57 (1926).
67
C. J. Gorter and R. de L. Kronig, “On the Theory of Absorption and Dispersion
in Paramagnetic and Dielectric Media,” Physica III, 9, 1009 –20 (1936).
68
H. A. Kramers, “La Diffusion de la Lumière par les Atomes,” Atti. Congr. Intern.
Fis. Como., 2, 545–57 (1927).
69
M. L. Bortz and R. H. French, “Quantitative, FFT-Based, Kramers Kronig
Analysis for Reflectance Data,” Appl. Spectrosc., 43 [8] 1498 –501 (1989).
70
D. Y. Smith, “Dispersion Theory, Sum Rules, and Their Application to the
Analysis of Optical Data”; pp. 35– 68 in Handbook of Optical Constants of Solids.
Edited by E. D. Palik. Academic Press, New York, 1985.
71
E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, “Self-Consistency and
Sum-Rule Tests in the Kramers–Kronig Analysis of Optical Data: Applications to
Aluminum,” Phys. Rev. B: Condens. Matter, 22, 1612–26 (1980).
72
R. F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope,
2nd ed.; p. 485. Plenum Press, New York, 1996.
73
A. D. Dorneich, R. H. French, H. Müllejans, S. Loughin, and M. Rühle,
“Quantitative Analysis of Valence Electron Energy-Loss Spectra of Aluminum
Nitride,” J. Microsc., 191 [3] 286 –96 (1998).
74
H. Müllejans and R. H. French, “Insights into the Electronic Structure of
Ceramics through Quantitative Analysis of Valence Electron Energy-Loss Spectroscopy (VEELS),” Microsc. Microanaly., 6, 297–306 (2000).
75
R. H. French and W. Y. Ching, “Ab Initio Calculation of Hamaker Constants of
Silicon Oxides and Silicon Nitrides Using Lifshitz Theory,“ unpublished work.
76
D. B. Hough and L. R. White, “The Calculation of Hamaker Constants from
Lifshitz Theory with Application to Wetting Phenomena,” Adv. Colloid Interface Sci.,
14, 3– 41 (1980).
77
E. D. Palik (Ed.), Handbook of Optical Constants of Solids, Vol. I, 1985; Vol II,
1991; Vol. III, 1998. Academic Press, New York.
78
F. Gervais, “Aluminum Oxide (Al2O3)”; pp. 761–75 in Handbook of Optical
Constants of Solids, Vol. II. Edited by E. D. Palik. Academic Press, New York, 1991.
79
W. J. Tropf and M. E. Thomas, “Aluminum Oxide (Al2O3) Revisited”; pp.
777– 87 in Handbook of Optical Constants of Solids, Vol. III. Edited by E. D. Palik.
Academic Press, New York, 1998.
80
N. G. Van Kampen, B. R. A. Nijboer, and K. Schram, “On the Macroscopic
Theory of van der Waals Forces,” Phys. Lett., 26A, 307–308 (1968).
81
D. C. Prieve and W. B. Russell, “Simplified Predictions of Hamaker Constants
from Lifshitz Theory,” J. Colloid Interface Sci., 125 [1] 1–13 (1988).
82
L. Bergström, A. Meurk, H. Arwin, and D. J. Rowecliffe, “Estimation of
Hamaker Constants of Ceramic Materials from Optical Data Using Lifshitz Theory,”
J. Am. Ceram. Soc., 79, 339 – 48 (1996).
83
L. Bergström, S. Stemme, T. Dahlfors, H. Arwin, and L. Ödberg, “Spectroscopic
Ellipsometry Characterisation and Estimation of the Hamaker Constant of Cellulose,”
Cellulose, 6, 1–13 (1999).
84
V. A. Parsegian and G. H. Weiss, “Spectroscopic Parameters for Computation of
van der Waals Forces,” J. Colloid Interface Sci., 81, 285– 89 (1981).
85
C. M. Roth and A. M. Lenhoff, “Improved Parametric Representation of Water
Dielectric Data for Lifshitz Theory Calculations,” J. Colloid Interface Sci., 179,
637–39 (1996).
86
R. M. Pashley, “The van der Waals Interaction for Liquid Water: A Comparison
of the Oscillator Model Approximation and Use of the Kramers–Kronig Equation
with Full Spectral Data,” J. Colloid Interface Sci., 62, 344 – 47 (1977).
87
H. D. Ackler, R. H. French, and Y. M. Chiang, “Comparison of Hamaker
Constants for Ceramic Systems with Intervening Vacuum or Water: From Force Laws
and Physical Properties,” J. Colloid Interface Sci., 179, 460 – 69 (1996).
88
M. R. Querry, D. M. Wieliczka, and D. J. Segelstein, “Water (H2O)”; see Ref. 78,
pp. 1059 –78.
89
O. V. Kopelevich, “Optical Properties of Pure Water in the 250 – 600 nm Range,”
Opt. Spektrosk., 41, 391–92 (1977).
90
J. Lenoble, and B. Saint-Guilly, “Sur l’absorption du Rayonnement Ultraviolet
par l’eau Distillée,” Compt. Rend., 240, 954 –55 (1955).
91
L. R. Painter, R. D. Birkhoff, and E. T. Arakawa, “Optical Measurements of
Liquid Water in the Vacuum Ultraviolet,” J. Chem. Phys., 51, 243–51 (1969).
2143
92
J. M. Heller Jr., R. N. Hamm, R. D. Birkhoff, and L. R. Painter, “Collective
Oscillation in Liquid Water,” J. Chem. Phys., 60, 3483– 86 (1974).
93
D. Tabor and R. H. S. Winterton, “The Direct Measurement of Normal and
Retarded van der Waals Forces,” Proc. R. Soc. London, A, 312, 435–50 (1969).
94
J. N. Israelachvili, Eqs. 11.13 and 11.14 in Intermolecular and Surface Forces,
2nd ed.; p. 184. Academic Press, London, U.K., 1992.
95
R. E. Johnson and R. H. Dettre, “Wetting of Low-Energy Surfaces”; pp. 1–73 in
Wettability, Surfactant Science Series, Vol. 49. Edited by J. C. Berg. Marcel Dekker,
New York, 1993.
96
J. N. Israelachvili, “van der Waals Force Contribution to Works of Adhesion and
Contact Angles on the Basis of Macroscopic Theory,” J. Chem. Soc., Faraday Trans.
2, 69, 1729 –38 (1974).
97
K. Autumn, Y. A. Liang, S. T. Hsieh, W. Zesch, W. P. Chan, T. W. Kenny, R.
Fearing, and R. I. Full, “Adhesive Force of a Single Gecko Foot-Hair,” Nature
(London), 405, 681– 85 (2000).
98
W. A. Zisman, “Relation of the Equilibrium Contact Angle to Liquid and Solid
Constitution,” Adv. Chem., 43, 1–51 (1964).
99
R. J. Good and L. A. Girifalco, “A Theory for Estimation of Surface and
Interfacial Energies, III, Estimation of Surface Energies of Solids from Contact Angle
Data,” J. Phys. Chem., 64, 561– 65 (1960).
100
D. R. Clarke and M. L. Gee, “Wetting of Surfaces and Grain Boundaries”; pp.
255–72 in Materials Interfaces. Chapman and Hall, London, U.K., 1992.
101
D. R. Clarke, “Possible Electrical Double-Layer Contribution to the Equilibrium
Thickness of Intergranular Glass Films in Polycrystalline Ceramics,” J. Am. Ceram.
Soc., 76, 1201–204 (1993).
102
D. R. Clarke, “Wetting of Grain Boundaries in Metals and Ceramics”; pp. 1– 8
in Intergranular and Interphase Boundaries in Materials, Materials Science Forum,
Vol. 294 –296. TransTech, Aedermannsdorf, Switzerland, 1999.
103
J. W. Cahn, “Critical Point Wetting,” J. Chem. Phys., 66 [8] 3667–72 (1977).
104
J. W. Cahn and J. E. Hilliard, “Free Energy of a Nonuniform System. I.
Interfacial Free Energy,” J. Chem. Phys., 28, 258 – 67 (1958).
105
S. Dietrich, “Fluid Interfaces: Wetting, Critical Adsorption, van der Waals Tails,
and the Concept of the Effective Interface Potential”; pp. 391– 423 in Phase
Transitions in Surface Films, Vol. 2. Edited by H. Taub, G. Torzo, H. J. Lauter, and
S. C. Fain Jr. Plenum Press, New York, 1991.
106
Y. Solomentsev and L. R. White, “Microscopic Drop Profiles and the Origins of
Line Tension,” J. Colloid Interface Sci., 218, 122–36 (1999).
107
M. Elbaum and S. G. Lipson, “How Does a Thin Wetted Film Dry Up?,” Phys.
Rev. Lett., 72, 3562– 65 (1994).
108
R. Yerushalmi-Rozen, J. Klein, and L. J. Fetters, “Suppression of Rupture in
Thin, Nonwetting Liquid Films,” Science (Washington), 263, 793–95 (1994).
109
S. Wu, Polymer Interface and Adhesion; pp. 87–94. Marcel Dekker, New York,
1982.
110
G. T. Dee and B. B. Sauer, “The Molecular Weight and Temperature Dependence of Polymer Surface Tension: Comparison of Experiment with Interface
Gradient Theory,” J. Colloid Interface Sci., 152, 85–103 (1992).
111
J. A. Lewis, “Colloidal Processing of Ceramics,” J. Am. Ceram. Soc., 83 [10]
(2000), in press.
112
J. Th. G. Overbeek and M. J. Sparnaay, “I. Classical Coagulation: London–van
der Waals Attraction Between Macroscopic Objects.” Discuss. Faraday Soc., 18,
13–25 (1954).
113
B. V. Derjaguin and L. Landau, “Theory of the Stability of Strongly Charged
Lyophobic Sols and of the Adhesion of Strongly Charged Particles in Solutions of
Electrolytes,” Acta Phys. Chim. U.R.S.S., 14, 633 (1941).
114
I. I. Abrikossova and B. V. Derjaguin, “Measurement of Molecular Attraction
Between Solid Bodies of Different Nature at Large Distances”; pp. 398 – 405 in
Proceedings of the Second International Congress on Surface Activity, Vol. 3
(London, U.K., 1957). Butterworths, London, U.K., 1957.
115
J. N Israelachvili, Intermolecular and Surface Forces, 2nd ed.; p. 169. Academic
Press, London, U.K., 1992.
116
R. G. Horn, D. R. Clarke, and M. T. Clarkson, “Direct Measurement of Surface
Forces Between Sapphire Crystals in Aqueous Solutions,” J. Mater. Res., 3, 413–16
(1988).
117
J. N. Israelachvili and G. E. Adams, “Measurement of Forces Between Two
Mica Surfaces in Aqueous Electrolyte Solutions in the Range 0 –100 nm,” J. Chem.
Soc., 74, 975–1002 (1978).
118
R. G. Horn and J. N. Israelachvili, “Direct Measurement of Forces due to
Solvent Structure,” Chem. Phys. Lett., 71, 192–94 (1980).
119
R. M. Pashley and J. N. Israelachvili, “Molecular Layering of Water in Thin
Films Between Mica Surfaces and Its Relation to Hydration Forces,” J. Colloid
Interface Sci., 101, 511–23 (1984).
120
G. Binnig, C. F. Quate, and Ch. Gerber, “Atomic Force Microscope,” Phys. Rev.
Lett., 56, 930 –33 (1985).
121
A. L. Weisenhorn, P. K. Hansma, T. R. Albrecht, and C. F. Quate, “Forces in
Atomic Force Microscopy in Air and Water,” Appl. Phys. Lett., 54, 2651–53 (1989).
122
W. A. Ducker, T. J. Senden, and R. M. Pashley, “Measurements of Forces in
Liquids Using a Force Microscope,” Langmuir, 8 [7] 1831–36 (1992).
123
T. J. Senden, C. J. Drummond, and P. Kékicheff, “Atomic Force Microscopy:
Imaging with Electrical Double Layer Interactions,” Langmuir, 10, 358 – 62 (1994).
124
U. Hartmann, “Theory of van der Waals Microscopy,” J. Vac. Sci. Technol. B,
9, 465– 69 (1991).
125
C. Argento and R. H. French, “Parametric Tip Model and Force–Distance
Relation for Hamaker Constant Determination from AFM,” J. Appl. Phys., 80,
6081–90 (1996).
126
W. A. Ducker and D. R. Clarke, “Controlled Modification of Silicon Nitride
Interactions in Water Via Zwitterionic Surfactant Adsorption,” Colloids Surf. A, 94,
275–92 (1994).
127
I. Larson, C. J. Drummond, D. Y. C. Chan, and F. Grieser, “Direct Force
Measurements Between TiO2 Surfaces,” J. Am. Chem. Soc., 115 [25] 11885–90
(1993).
2144
Journal of the American Ceramic Society—French
128
S. Biggs and P. Mulvaney, “Measurement of the Forces Between Gold Surfaces
in Water by Atomic Force Microscopy,” J. Chem. Phys., 100, 8501–505 (1994).
129
W. A. Ducker, Z. Xu, D. R. Clarke, and J. N. Israelachvili, “Forces Between
Alumina Surfaces in Salt Solutions: Non-DLVO Forces and the Implications for
Colloidal Processing,” J. Am. Ceram. Soc., 77, 437– 43 (1994).
130
M. S. Hook, P. G. Hartley, and P. J. Thistlethwaite, “Fabrication and Characterization of Spherical Zirconia Particles for Direct Force Measurements Using the
Atomic Force Microscope,” Langmuir, 15, 6220 –25 (1999).
131
H. G. Pedersen and L. Bergström, “Forces Measured Between Zirconia Surfaces
in Poly(acrylic acid) Solutions,” J. Am. Ceram. Soc., 82, 1137– 45 (1999).
132
H. Bluhm, T. Inoue, and M. Salmeron, “Friction of Ice Measured Using Lateral
Force Microscopy,” Phys. Rev. B: Condens. Matter, 61, 7760 – 65 (2000).
133
G. I. Barenblatt, “The Mathematical Theory of Equilibrium Cracks in Brittle
Fracture,” Adv. Appl. Mech., 7, 55–129 (1962).
134
A. Jagota, C. Argento, and S. Mazur, “Growth of Adhesive Contacts for
Maxwell Viscoelastic Spheres,” J. Appl. Phys., 83 [1] 250 –59 (1998).
135
C. Argento, A. Jagota, and W. C. Carter, “Surface Formulation for Molecular
Interactions of Macroscopic Bodies,” J. Mech. Phys. Solids, 45 [7] 1161– 83 (1997).
136
A. W. Adamson, Physical Chemistry of Surfaces, 3rd ed.; Ch. 1. Wiley, New
York, 1976.
137
V. B. Bezerra, G. L. Klimchitskaya, and C. Romero, “Casimir Force Between a
Flat Plate and a Spherical Lens: Applications to the Results of a New Experiment,”
Modern Phys. Lett. A, 12, 2613–22 (1997).
138
B. V. Derjaguin, “Untersuchungen über die Reibung und Adhesion, IV. Theorie
des Anhaftens kleiner Teilchen,” Kolloid-Z., 69, 155– 64 (1934).
139
A. Jagota and C. Argento, “An Intersurface Stress Tensor,” J. Colloid Interface
Sci., 191, 326 –36 (1997).
140
R. M. McMeeking, “A Maxwell Stress for Material Interactions,” J. Colloid
Interface Sci., 199, 187–96 (1998).
141
J. D. Jackson, Classical Electrodynamics, 2nd ed. Wiley, New York, 1975.
142
M. F. Ashby, “A First Report on Sintering Diagrams,” Acta Metall., 22, 275– 89
(1974).
143
F. B. Swinkels and M. F. Ashby, “A Second Report on Sintering Diagrams,”
Acta Metall., 29, 259 – 81 (1981).
144
K. L. Johnson, K. Kendall, and A. D. Roberts, “Surface Energy and the Contact
of Elastic Solids,” Proc. R. Soc. London, A, 324, 301–13 (1971).
145
J. Eggers, “Coalescence of Spheres by Surface Diffusion,” Phys. Rev. Lett., 80,
2634 –37 (1998).
146
G. C. Kuczynski, “Study of the Sintering of Glass,” J. Appl. Phys., 20, 1160 – 63
(1949).
147
C. Herring, “Effect of Change of Scale on Sintering Phenomena,” J. Appl. Phys.,
21, 301–303 (1950).
148
S. Mazur and D. J. Plazek, “Viscoelastic Effects in the Coalescence of Polymer
Particles,” Prog. Org. Coatings, 24, 225–36 (1994).
149
R. M. Cannon and L. Esposito, “High-Temperature Colloidal Behavior: Particles in Liquid Silicates,” Z. Metallkd., 90, 1002–15 (1999).
150
L. Esposito, A. Bellosi, and S. Landi, “Interfacial Forces in Si3N4- and
SiC-Based Systems and Their Influence on the Joining Process,” J. Am. Ceram. Soc.,
82, 3597– 604 (1999).
151
J. White, “Microstructure and Grain Growth in Ceramics in the Presence of a
Liquid Phase”; pp. 81–108 in Sintering and Related Phenomena. Edited by G. C.
Kuczynski. Plenum Press, New York, 1973.
152
W. A. Kaysser, M. Sprissler, C. A. Handwerker, and J. E. Blendell, “Effect of
a Liquid Phase on the Morphology of Grain Growth in Alumina,” J. Am. Ceram. Soc.,
70 [5] 339 – 43 (1987).
153
M. J. Hoffmann, “Analysis of Microstructure Development and Mechanical
Properties of Si3N4 Ceramics”; pp. 59 –72 in Tailoring of Mechanical Properties of
Si3N4 Ceramics. Edited by M. J. Hoffmann and G. Petzow. Kluwer, Dordrecht, The
Netherlands, 1994.
154
W. D. Kingery, “Distribution and Influence of Minor Constituents on Ceramic
Formulations”; pp. 281–94 in Ceramic Microstructures ’86: Role of Interfaces.
Edited by J. A. Pask and A. G. Evans. Plenum Press, New York, 1987.
155
H.-J. Kleebe, “Structure and Chemistry of Interfaces in Si3N4 Materials Studied
by Transmission Electron Microscopy,” J. Ceram. Soc. Jpn., 105 [6] 453–75 (1997).
156
R. Raj, “Fundamental Research in Structural Ceramics for Service Near
2000°C,” J. Am. Ceram. Soc., 76 [9] 2147–74 (1993).
157
S. M. Wiederhorn, B. J. Hockey, and J. D. French, “Mechanisms of Deformation
of Silicon Nitride and Silicon Carbide at High Temperature,” J. Eur. Ceram. Soc., 19,
2273– 84 (1999).
158
P. F. Becher, E. Y. Sun, K. P. Plucknett, K. B. Alexander, C.-H. Hsueh, H-T.
Lin, S. B. Waters, and C. G. Westmoreland, “Microstructural Design of Silicon
Nitride with Improved Fracture Toughness: I, Effects of Grain Shape and Size,”
J. Am. Ceram. Soc., 81 [11] 2821–30 (1998).
159
E. Y. Sun, P. F. Becher, K. P. Plucknett, C.-H. Hsueh, K. B. Alexander, S. B.
Waters, K. Hirao, and M. E. Brito, “Microstructural Design of Silicon Nitride with
Improved Fracture Toughness: II, Effects of Yttria and Alumina Additions,” J. Am.
Ceram. Soc., 81 [11] 2831– 40 (1998).
160
(a)N. P. Padture, “In Situ-Toughened Silicon Carbide,“ J. Am. Ceram. Soc., 77
[2] 519 –23 (1994). (b) K. M. Knowles and S. Turan, “The Dependence of
Equilibrium Film Thickness on Grain Orientation at Interphase Boundaries in
Ceramic–Ceramic Composites,” Ultramicroscopy, 83, 245–59 (2000).
161
J. J. Cao, W. J. Moberly-Chan, L. C. DeJonghe, C. J. Gilbert, and R. O. Ritchie,
“In Situ-Toughened Silicon Carbide with Al-B-C Additions,“ J. Am. Ceram. Soc., 79
[2] 461– 69 (1996).
162
K. M. Knowles, S. Turan, A. Kumar, S.-J. Chen, and W. J. Clegg, “Transmission
Electron Microscopy of Interfaces in Structural Ceramic Composites,” J. Microsc.,
196 [Part 2] 194 –202 (1999).
163
Y.-M. Chiang, L. A. Silverman, R. H. French, and R. M. Cannon, “Thin Glass
Film between Ultrafine Conductor Particles in Thick-Film Resistors,” J. Am. Ceram.
Soc., 77 [5] 1143–52 (1994).
Vol. 83, No. 9
164
F. Greuter, “Electrically Active Interfaces in ZnO Varistor,” Solid State Ionics,
75 [1] 67–78 (1995).
165
R. Ramesh, S. M. Green, and G. Thomas, “Microstructure Property Relations in
the Bi(Pb)-Sr-Ca-Cu-O Ceramic Superconductors”; pp. 363– 403 in Studies of
High-Temperature Superconductors, Advances in Research and Applications, Vol. 5.
Nova Science Publications, Commack, NY, 1990.
166
H. Gu, R. M. Cannon, and M. Rühle, “Composition and Chemical Width of
Ultra-Thin Intergranular Amorphous Film in Silicon Nitride,” J. Mater. Res., 13 [2]
476 – 87 (1998).
167
H. Gu, X. Pan, R. M. Cannon, and M. Rühle, “Dopant Distribution in
Grain-Boundary Films in Calcia-Doped Silicon Nitride Ceramics,” J. Am. Ceram.
Soc., 81 [12] 3125–35 (1998).
168
H. Gu, R. M. Cannon, H. J. Seifert, and M. J. Hoffmann, “Solubility of Si3N4
in Liquid SiO2,” J. Am. Ceram. Soc., in review.
169
I. Tanaka, H. Adachi, T. Nakayasu, and T. Yamada, “Local Chemical Bonding
at Grain Boundary of Si3N4 Ceramics”; pp. 23–34 in Ceramic Microstructure:
Control at the Atomic Level. Edited by A. P. Tomsia and A. Glaeser. Plenum Press,
New York, 1998.
170
Y.-M. Chiang, J.-R. Lee, and H. Wang, “Microstructure and Intergranular Phase
Distribution in Bi2O3-Doped ZnO”; see Ref. 172, pp. 131– 47.
171
R. Brydson, S.-C. Chen, F. L. Riley, S. J. Milne, X. Pan, and M. Rühle,
“Microstructure and Chemistry of Intergranular Glassy Films in Liquid-PhaseSintered Alumina,” J. Am. Ceram. Soc., 81 [2] 369 –79 (1998).
172
G. H. Gessinger, H. F. Fischmeister, and H. L. Lukas, “A Model for
Second-Stage Liquid-Phase Sintering with a Partially Wetting Liquid,” Acta Metall.,
21 [5] 715–24 (1973).
173
H.-H. Park, S.-J. L. Kang, and D. N. Yoon, “An Analysis of the Surface Menisci
in a Mixture of Liquid and Deformable Grains,” Metall. Trans., 17A [2] 325–30
(1986).
174
T. M. Shaw, “Model for the Effect of Powder Packing on the Driving Force for
Liquid-Phase Sintering,” J. Am. Ceram. Soc., 76 [3] 664 –70 (1993).
175
J. Luo, H. Wang, and Y.-M. Chiang, “Origin of Solid-State Activated Sintering
in Bi2O3-Doped ZnO,” J. Am. Ceram. Soc., 82 [4] 916 –20 (1999).
176
D. R. Clarke and G. Thomas, “Grain Boundary Phases in a Hot-Pressed MgO
Fluxed Silicon Nitride,” J. Am. Ceram. Soc., 60 [11–12] 491–95 (1977).
177
L. K. V. Lou, T. E. Mitchell, and A. H. Heuer, “Impurity Phases in Hot-Pressed
Si3N4,” J. Am. Ceram. Soc., 61 [9 –10] 392–96 (1978).
178
O. L. Krivanek, T. M. Shaw, and G. Thomas, “The Microstructure and
Distribution of Impurities in Hot-Pressed and Sintered Silicon Nitrides,” J. Am.
Ceram. Soc., 62 [11–12] 585–90 (1979).
179
H.-J. Kleebe, M. K. Cinibulk, R. M. Cannon, and M. Rühle, “Statistical Analysis
of the Intergranular Film Thickness in Silicon Nitride Ceramics,” J. Am. Ceram. Soc.,
76 [8] 1969 –77 (1993).
180
I. Tanaka, H.-J. Kleebe, M. K. Cinibulk, J. Bruley, D. R. Clarke, and M. Rühle,
“Calcium Concentration Dependence of the Intergranular Film Thickness in Silicon
Nitride,” J. Am. Ceram. Soc., 77, 911–14 (1994).
181
C.-M. Wang, X. Pan, M. J. Hoffmann, R. M. Cannon, and M. Rühle, “Grain
Boundary Films in Rare-Earth-Glass-Based Silicon Nitride,” J. Am. Ceram. Soc., 79
[3] 788 –92 (1996).
182
D. R. Clarke, “On the Equilibrium Thickness of Intergranular Glassy Phases in
Ceramic Materials,” J. Am. Ceram. Soc., 70 [1] 15–22 (1987).
183
D. R. Clarke, T. M. Shaw, A. Philipse, and R. G. Horn, “Possible Electrical
Double-Layer Contribution to the Equilibrium Thickness of Intergranular Glass
Phases in Polycrystalline Ceramics,” J. Am. Ceram. Soc., 76 [5] 1201–204 (1994).
184
H. Wang and Y.-M. Chiang, “Thermodynamic Stability of Intergranular
Amorphous Films in Bismuth-Doped Zinc Oxide,” J. Am. Ceram. Soc., 81 [1] 89 –96
(1998).
185
T. M. Shaw and P. R. Duncombe, “Forces between Aluminum Oxide Grains in
a Silicate Melt and Their Effect on Grain Boundary Wetting,” J. Am. Ceram. Soc., 74
[10] 2495–505 (1991).
186
P. L. Flaitz and J. A. Pask, “Penetration of Polycrystalline Alumina by Glass at
High Temperatures,” J. Am. Ceram. Soc., 70 [7] 449 –55 (1987).
187
J.-R. Lee and Y.-M. Chiang, “Bi Segregation at ZnO Grain Boundaries in
Equilibrium with Bi2O3–ZnO Liquid,” Solid State Ionics, 75, 79 – 88 (1995).
188
L. Esposito, E. Saiz, A. P. Tomsia, and R. M. Cannon, “High-Temperature
Colloidal Processing for Glass/Metal and Glass/Ceramic FGM’s”; pp. 503–12 in
Ceramic Microstructure: Control at the Atomic Level. Edited by A. P. Tomsia and A.
Glaeser. Plenum Press, New York, 1998.
189
T. H. Courtney, “Densification and Structural Development in Liquid Phase
Sintering,” Metall. Trans A., 15A, 1065–74 (1984).
190
J. Jitcharoen, N. Padture, A. E. Giannakopoulos, and S. Suresh, “Hertzian-Crack
Suppression in Ceramics with Elastic-Modulus Graded Surfaces,” J. Am. Ceram.
Soc., 81 [9] 2301–308 (1998).
191
D. C. Pender, “Ceramics with Graded Surfaces for Contact Damage Resistance”;
Ph.D. Dissertation. University of Connecticut, Storrs, CT, 1999.
192
L. Esposito and A. Bellosi, “The Role of Intergranular Phases in Si3N4 and SiC
Direct Joining”; pp. 211–22 in Interfacial Science in Ceramic Joining. Edited by A.
Bellosi, T. Kosmac, and A. P. Tomsia. Kluwer, Boston, MA, 1998.
193
M. Gopal, L. C. DeJonghe, and G. Thomas, “Silicon Nitride: From Sintering to
Joining,” Acta Mater., 46 [7] 2401– 405 (1998).
194
Y. K. Simpson, C. B. Carter, K. J. Morrissey, P. Angelini, and J. Bentley, “The
Identification of Thin Amorphous Films at Grain Boundaries in Al2O3,” J. Mater.
Sci., 21, 2689 –96 (1986).
195
M. Rühle, N. Claussen, and A. H. Heuer, “Microstructural Studies of Y2O3Containing Tetragonal ZrO2 Polycrystals (Y-TZP),” Adv. Ceram., 12, 352–70 (1984).
196
L. K. V. Lou, T. E. Mitchell, and A. H. Heuer, “Impurity Phases in Hot-Pressed
Si3N4,” J. Am. Ceram. Soc., 61, 392–96 (1978).
197
D. L. Callahan and G. Thomas, “Impurity Distribution in Polycrystalline
Aluminum Nitride Ceramics,” J. Am. Ceram. Soc., 73, 2167–70 (1990).
September 2000
Origins and Applications of London Dispersion Forces and Hamaker Constants in Ceramics
198
F. F. Lange and D. R. Clarke, “Morphological Changes of an Intergranular Thin
Film in a Polycrystalline Spinel,” J. Am. Ceram. Soc., 65, 502–506 (1982).
199
S. C. Hansen and D. S. Phillips, “Grain-Boundary Microstructures in a
Liquid-Phase-Sintered Alumina (␣-Al2O3),” Philos. Mag. A, A47 [2] 209 –34 (1983).
200
D. W. Susnitzky and C. B. Carter, “Structure of Alumina Grain Boundaries
Prepared with and without a Thin Amorphous Intergranular Film,” J. Am. Ceram.
Soc., 73 [8] 2485–93 (1990).
201
C. A. Powell-Dogan and A. H. Heuer, “Microstructure of 96% Alumina
Ceramics: I, Characterization of the As-Sintered Materials,” J. Am. Ceram. Soc., 73
[12] 3670 –76 (1990).
202
D.-Y. Kim, S. M. Wiederhorn, B. J. Hockey, J. E. Blendell, and C. A.
Handwerker, “Stability and Surface Energies of Wetted Grain Boundaries in Aluminum Oxide,” J. Am. Ceram. Soc., 77 [2] 444 –53 (1994).
203
J. E. Blendell, W. C. Carter, and C. A. Handwerker, “Faceting and Wetting
Transitions of Anisotropic Interfaces and Grain Boundaries,” J. Am. Ceram. Soc., 82
[7] 1889 –900 (1999).
204
P. L. Flaitz and J. A. Pask, “Penetration of Polycrystalline Alumina by Glass at
High Temperatures,” J. Am. Ceram. Soc., 70 [7] 449 –55 (1987).
205
T. M. Shaw and P. R. Duncombe, “Forces between Aluminum Oxide Grains in
a Silicate Melt and Their Effect on Grain Boundary Wetting,” J. Am. Ceram. Soc., 74
[10] 2495–505 (1991).
206
S. Ramamurthy, M. P. Mallamaci, C. M. Zimmerman, C. B. Carter, P. R.
Duncombe, and T. M. Shaw, “Microstructure of Polycrystalline MgO Penetrated by
a Silicate Liquid,” JMSA ISSN, 2 [3] 113–28 (1996).
207
F. Ernst, O. Kienzle, and M. Rühle, “Structure and Composition of Grain
Boundaries in Ceramics,” J. Eur. Ceram. Soc., 19, 665–73 (1999).
208
O. Kienzle, “Atomistische Struktur und Chemische Zusammensetzung innerer
Grenzflächen von Strontiumtitanat”; Ph.D. Dissertation. University of Stuttgart and
Max-Planck Institute for Metals Research, Stuttgart, Germany, 1999.
209
J. P. Gambino, W. D. Kingery, G. E. Pike, and H. R. Phillip, “Effect of Heat
Treatments on the Wetting Behavior of Bismuth-Rich Intergranular Phases in
ZnO:Bi:Co Varistors,” J. Am. Ceram. Soc., 72, 642– 45 (1989).
210
E. Olsson and G. L. Dunlop, “The Effect of Bi2O3 Content on the Microstructure
and Electrical Properties of ZnO Varistor Materials,” J. Appl. Phys., 66, 4317–24
(1989).
211
M. Fujimoto and W. D. Kingery, “Microstructures of SrTiO3 Internal Boundary
Layer Capacitors During and After Processing and the Resultant Electrical Properties,” J. Am. Ceram. Soc., 68 [4] 169 –73 (1985).
212
M. Kramer, M. J. Hoffmann, and G. Petzow, “Grain Growth Studies of Silicon
Nitride Dispersed in an Oxynitride Glass,” J. Am. Ceram. Soc., 76, 2778 – 84 (1993).
213
I. Tanaka, J. Bruley, H. Gu, M. J. Hoffmann, H. J. Kleebe, R. M. Cannon, D. R.
Clarke, and M. Rühle; pp. 275– 89 in Tailoring of Mechanical Properties of Si3N4
Ceramics. Kluwer Academic Publishers, Stuttgart, Germany, 1994.
214
J.-R. Lee, Y.-M. Chiang, and G. Ceder, “Pressure–Thermodynamic Study of
Grain Boundaries: Bi Segregation in ZnO,” Acta Mater., 45[3] 1247–57 (1997).
215
H. Wang and Y.-M. Chiang, “Thermodynamic Stability of Intergranular
Amorphous Films in Bi-Doped ZnO,” J. Am. Ceram. Soc., 81[1] 89 –96 (1998).
216
Y.-M. Chiang, H. Wang, and J.-R. Lee, “HREM and STEM of Intergranular
Films at Zinc Oxide Varistor Grain Boundaries,” J. Microsc., 191 [3] 275– 85 (1998).
2145
217
J. Luo, H. Wang, and Y.-M. Chiang, “Origin of Solid-State Activated Sintering
in ZnO–Bi2O3,” J. Am. Ceram. Soc., 82 [4] 916 –20 (1999).
218
T. D. Chen, J.-R. Lee, H. L. Tuller, and Y.-M. Chiang, “Dopant and
Heat-Treatment Effects on the Electrical Properties of Polycrystalline ZnO”; pp.
295–300 in Electrically Based Microstructural Characterization, Materials Research
Society Symposium Proceedings, Vol. 41. Materials Research Society, Pittsburgh,
PA, 1996.
219
J. Luo and Y.-M. Chiang, “Equilibrium Thickness Amorphous Films on {112៮ 0}
Surfaces of Bi2O3-Doped ZnO, J. Eur. Ceram. Soc., 19, 697–701 (1999).
220
J. Juo and Y.-M. Chiang, “Existence and Stability of Nanometer-Thick Amorphous Films on Oxide Surfaces,” Acta Mater., in review.
221
V. A. Parsegian and G. H. Weiss, “Dielectric Anisotropy and the van der Waals
Interaction Between Bulk Media,” J. Adhes., 3, 259 – 67 (1972).
222
V. A. Parsegian and G. H. Weiss, “On van der Waals Interactions Between
Macroscopic Bodies Having Inhomogeneous Dielectric Susceptibilities,” J. Colloid
Interface Sci., 40, 35– 41 (1972).
223
H. D. Ackler, W. C. Carter, and Y.-M. Chiang, unpublished work.
224
M. Bobeth, D. R. Clarke, and W. Pompe, “A Diffuse Interface Description of
Intergranular Films in Polycrystalline Ceramics,” J. Am. Ceram. Soc., 82, 1537– 46
(1999).
225
J. A. Warren, W. C. Carter, and R. Kobayashi, “A Phase Field Model of the
Impingement of Solidifying Particles,” Physica A (Amsterdam), 14, 8 (1998).
226
D. A. Muller, T. Sorsch, S. Moccio, F. H. Baumann, K. Evans-Lutterodt, and G.
Timp, “The Electronic Structure at the Atomic Scale of Ultrathin Gate Oxides,”
Nature (London), 399, 758 – 61 (1999).
227
N. A. Modine, G. Zumbach, and E. Kaxiras, “Adaptive-Coordinate Real-Space
Electronic-Structure Calculations for Atoms, Molecules, and Solids,” Phys. Rev. B:
Condens. Matter, 55, 10289 –301 (1997).
228
J. S. Wettlaufer, “Impurity Effects in the Premelting of Ice,” Phys. Rev. Lett., 82,
2516 –19 (1999).
229
R. Bar-Ziv and S. A. Safran, “Surface Melting of Ice Induced by Hydrocarbon
Films,” Langmuir, 9, 2786 – 88 (1993).
230
J. Brandrup and E. H. Immergut, The Polymer Handbook, 3rd ed. Wiley
Interscience, New York, 1989.
231
A. Souter, DuPont Packaging and Industrial Polymers, private communication.
232
C. A. Harper, Handbook of Plastics and Elastomers. McGraw-Hill, New York,
1975.
233
(a)S. Wu, Polymer Interface and Adhesion; pp. 87–94. Marcel Dekker, New
York, 1982. (b)S. Wu, “Surface and Interfacial Tensions of Polymers, Oligomers,
Plasticizers, and Organic Pigments”; pp. 411–34 in Polymer Handbook, 3rd ed. Wiley
Interscience, New York, 1989.
234
R. H. French, C. Scheu, G. Duscher, H. Müllejans, M. J. Hoffmann, and R. M.
Cannon, “Interfacial Electronic Structure and Full Spectral Hamaker Constants of
Si3N4 Intergranular Films from VUV and SR-VEEL Spectroscopy,” Mater. Res. Soc.
Symp. Proc., 357, 243–58 (1995).
235
H. D. Ackler and Y.-M. Chiang, “Effect of Initial Microstructure on Final
Intergranular Phase Distributions in Liquid-Phase-Sintered Ceramics,” J. Am. Ceram.
Soc., 82 [1] 183– 89 (1999).
䡺
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Journal of the American Ceramic Society—French
Vol. 83, No. 9
Roger H. French is a Senior Research Associate, Materials Science and Engineering,
Central Research and Development, DuPont Company, Wilmington, DE, and Adjunct
Professor, Materials Science Department, University of Pennsylvania, Philadelphia. He
received his B.S. in materials science from Cornell University and his Ph.D. from
Massachusetts Institute of Technology under R. L. Coble. French’s research is in optical
properties and electronic structure of ceramic, optical, and polymeric materials, and in
materials and microstructure design. These studies are pursued using vacuum ultraviolet
spectroscopy, spectroscopic ellipsometry, valence-electron energy-loss spectroscopy, and
computational optics. His current work is in the origins and applications of London dispersion
forces and the electronic structure and wetting of interfaces and intergranular films. French
studies near-field optics and scattering by particulate dispersions and complex microstructures using computational solutions to Maxwell’s equations. Work on new materials for
optical lithography for integrated chip fabrication at 365, 248, 193, and 157 nm wavelengths
has produced attenuating phase shift photomasks, photomask pellicles, and semiconductor
photoresists. His work has produced nine patents and filings and more than 85 published
papers. French is a Director of The American Ceramic Society, past chair of the Basic Science
Division, and an Associate Editor of the Journal of the American Ceramic Society. He has
served as co-chair for the Basic Science Division Program Committee and Gordon
Conference on Solid-State Studies in Ceramics. French is a Fulrath awardee and a Fellow of
The American Ceramic Society.